Maths in Focus Chapter 11

11 Probability TERMINOLOGY Arrangements: Different ways of organising objects Combinations: Arrangements of objects without replacement or repetition when order is not important Complement: The complement of an event E is when the event E does not occur Equally likely outcomes: Each outcome has the same chance of occurring Factorial: A factorial is the product of n consecutive positive integers from n down to 1. For example 6! = 6 x 5 x 4 x 3 x 2 x 1 Fundamental counting principle: If one event can occur in p ways and a second independent event can occur in q ways, then the two successive events can occur in p x q different ways Independent events: Events are independent if the result of one event does not influence the outcome of another event. There is no overlap between the events Multi-stage events: A series of successive independent events Mutually exclusive results: Two events with the same sample space that cannot both occur at the same time Non-mutually exclusive results: Two events with the same sample space that can occur at the same time i.e. there is some overlap

Ordered selections: Selections that are taken in a particular position or order. Permutations: The arrangement of objects without replacement or repetition when order is important Probability tree: A diagram that uses branches to show multi-stage events and sets out the probability on each branch with the sample space listed at the right of each branch Random experiments: Experiments that are made with no pattern or order where each outcome is equally likely to occur Sample space: The set of all possible outcomes in an event or series of events Tree diagram: A diagram that uses branches to show multi-stage events where the probabilities on each branch are equal Unordered selections: Selections that are made when the order of arrangements is not important or relevant Venn diagram: A special diagram to show the sample space for non-mutually exclusive events using circles for each event drawn inside a rectangle which represents the sample space

Chapter 11 Probability

Introduction Probability is the study of how likely it is that something will happen.

It is used to make predictions in different areas, ranging from gambling to determining the rate of insurance premiums. For example an actuary looking at death-rate statistics can estimate the probable age to which someone will live, and set life insurance premiums accordingly. Another example of where probability is useful is in biology. The probability of certain diseases or genetic defects can be calculated in high-risk families. Probability is also closely related to statistics and data analysis, as well as games of chance such as card games, tossing coins, backgammon and so on. It is also relevant to buying raffle and lottery tickets. In this chapter, you will revise simple probability that you have learnt in previous years and extend this to more complex probability involving multi-stage events.

DID YOU KNOW? Girolamo Cardano (1501–76) was a doctor and mathematician who developed the first theory of probability. He was a great gambler, and he wrote De Ludo Aleae (‘On Games of Chance’). This work was largely ignored, and it is said that the first book on probability was written by Christiaan Huygens (1629–95). The main study of probability was done by Blaise Pascal (1623–62), whom you have already heard about, and Pierre de Fermat (1601−65). Pascal developed the ‘arithmetical triangle‘ you studied in the last chapter. Pascal’s triangle has properties that are applicable to probability as well.

Simple Probability Mutually exclusive events Mutually exclusive events means that if one event occurs, the other cannot. For example, when rolling a die, a 6 cannot occur at the same time as a 2. We can measure probability in theory as long as the events are random, independent and equally likely to happen. However, even then, probability only gives us an approximate idea of the likelihood of certain events happening. For example, in Lotto draws, there is a machine that draws out the balls at random and a panel of supervisors checks that this happens properly. Each ball is independent of the others and is equally likely to be drawn out.

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In a horse race, it is difficult to measure probability as the horses are not all equally likely to win. Other factors such as ability, training, experience and weight of the jockey all affect it. The likelihood of any one horse winning is not random.

Class Discussion Discuss these statements: 1. The probability of one particular football team winning a 1 competition is as there are 16 teams. 16 2. The probability of Tiger Woods winning the US Open golf 1 tournament is if there are 78 players in the tournament. 78 3. A coin came up tails 8 times in a row. So there is a greater chance that the next time it will come up heads.

The probability of an event E happening, P(E), is given by the number of ways the event can occur, n(E), compared with the total number of outcomes possible n(S) (called the sample space) P (E) =

n (E) n (S)

If P(E) = 0 the event is impossible If P(E) = 1 the event is certain (it has to happen) 0 ≤ P(E) ≤ 1 The sum of all (mutually exclusive) probabilities is 1

Examples 1. A container holds 5 blue, 3 white and 7 yellow marbles. If one marble is selected at random, find the probability of getting (a) a white marble (b) a white or blue marble (c) a yellow, white or blue marble (d) a red marble.

Chapter 11 Probability

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Solution The sample space, or total number of marbles is 5 + 3 + 7 or 15. 3 15 1 = 5

(a) P (W) =

3+5 15 8 = 15

(b) P (W or B) =

7+3+5 15 15 = 15 =1

(c) P (Y or W or B) =

0 15 =0

(d) P (R) =

Getting a red marble is impossible!

2. The probability that a traffic light will turn green as a car approaches 5 it is estimated to be . A taxi goes through 192 intersections where there 12 are traffic lights. How many of these would be expected to turn green as the taxi approached?

Solution It is expected that

5 of the traffic lights would turn green. 12

5 ´ 192 = 80 12 So it would be expected that 80 traffic lights would turn green as the taxi approached.

11.1 Exercises 1. Peter is in a class of 30 students. If one student is chosen at random to make a speech, find the probability that the student chosen will be Peter.

2. A pack of cards contains 52 different cards, one of which is the ace of diamonds. If one card is chosen at random, find the probability that it will be the ace of diamonds.

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‘Die’ is singular for ‘dice’.

3. There are 6 different newspapers sold at the local newsagent each day. Kim sends her little brother to buy her a newspaper one morning but forgets to tell him which one. What is the probability that he will buy the correct newspaper? 4. A raffle is held in which 200 tickets are sold. If I buy 5 tickets, what is the probability of my winning the prize in the raffle? 5. In a lottery, 200 000 tickets are sold. If Lucia buys 10 tickets, what is the probability of her winning first prize? 6. A bag contains 6 red balls and 8 white balls. If one ball is drawn out of the bag at random, find the probability that it will be (a) white (b) red. 7. A shoe shop orders in 20 pairs of black, 14 pairs of navy and 3 pairs of brown school shoes. If the boxes are all mixed up, find the probability that one box selected at random will contain brown shoes. 8. Abdul has 12 CDs in his car glovebox. The labels have become mixed up. If he chooses one of the CDs at random, find the probability that it is his favourite CD. 9. A bag contains 5 black marbles, 4 yellow marbles and 11 green marbles. Find the probability of drawing 1 marble out at random and getting (a) a green marble (b) a yellow or a green marble.

10. A die is thrown. Calculate the probability of throwing (a) a 6 (b) an even number (c) a number less than 3. 11. A book has 124 pages. If the book is opened at any page at random, find the probability of the page number being (a) either 80 or 90 (b) a multiple of 10 (c) an odd number (d) less than 100. 12. In the game of pool, there are 15 balls, each with the number 1 to 15 on it. In Kelly Pool, each person chooses a number at random to determine which ball to sink. If Tracey chooses a number, find the probability that her ball will be (a) an odd number (b) a number less than 8 (c) the 8 ball. 13. At a school dance, each girl is given a ticket with a boy’s name on it. The girl then must dance with that boy for the next dance. If the tickets are given out at random and there are 50 boys at the dance, what is the probability that Jill will get to dance with her boyfriend? 14. Find the probability of a coin coming up heads when tossed. If the coin is double-headed, find the probability of tossing a head.

Chapter 11 Probability

15. In a bag of caramels, there are 21 with red wrappers and 23 with blue wrappers. If Leila chooses a caramel at random from the bag, find the probability that she will choose one with a blue wrapper. 16. A student is chosen at random to write about his/her favourite sport. If 12 students like tennis best, 7 prefer soccer, 3 prefer squash, 5 prefer basketball and 4 prefer swimming, find the probability that the student chosen will write about (a) soccer (b) squash or swimming (c) tennis. 17. A school has 875 students. If 5 students are chosen at random to help show some visitors around, find the probability that a particular student will be chosen. 18. A box containing a light globe 1 has a probability of holding 20 a defective globe. If 160 boxes are checked, how many would be expected to be defective? 19. There are 29 red, 17 blue, 21 yellow and 19 green chocolate beans in a packet. If Kate chooses one at random, find the probability that it will be red or yellow. 20. The probability of breeding a 2 white budgerigar is . If Mr Seed 9 breeds 153 budgerigars over the year, how many would be expected to be white?

21. A biased coin is weighted so that heads comes up twice as often as tails. Find the probability of tossing a tail. 22. A die has the centre dot painted white on the 5 so that it appears as a 4. Find the probability of throwing (a) a 2 (b) a 4 (c) a number less than 5. 23. Discuss these statements. (a) The probability of one particular horse winning the 1 Melbourne cup is if there are 20 20 horses in the race. (b) The probability of Greg Norman winning a masters golf 1 tournament is if there are 15 15 players in the tournament. (c) A coin came up tails 8 times in a row. So the next toss must be a head. (d) A family has three sons. There is more chance of getting a daughter next time. (e) The probability of a Holden winning the car race at Bathurst 6 as there are this year is 47 6 Holdens in the race and 47 cars altogether.

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Complementary events When we find the probabilities of events, the total of all the possible events will always add up to 1.

Example A ball is chosen at random from a bag containing 5 blue, 3 red and 7 yellow balls. The probabilities are as follows: 5 P (blue) = 15 3 P (red) = 15 7 P (yellow) = 15 3 5 7 Total probability = + + 15 15 15 15 = 15 =1

The complement of an event happening is the event not happening. That Eh is, the complement of P(E) is P(not E). We can write this as P ^ L

Example A die is thrown. Find the probability of (a) throwing a 6 (b) not throwing a 6.

Solution (a) P ] 6 g =

1 6

(b)  P (not 6) = P (1, 2, 3, 4, or 5) 5 = 6

In general, P (E) + P (L E) = 1 or P (E) = 1 − P (L E)

Chapter 11 Probability

Proof Let e be the number of ways E can happen out of a total of n events. Then the number of ways E will not happen is n − e. e Then P (E) = n n−e P (L E) = n n e =n −n e = 1− n = 1 − P (E)

Examples 1 . What is the probability of 1. The probability of a win in a raffle is 350 losing?

Solution P (lose) = 1 − P ] win g 1 = 1− 350 349 = 350 2. The probability of a tree surviving a fire is 72%. Find the probability of the tree failing to survive a fire.

Solution P ^ failing to survive h = 1 − P ^ surviving h = 100% − 72% = 28%

11.2 Exercises 1. The probability of a bus arriving 18 . What on time is estimated at 33 is the probability that the bus will not arrive on time? 2. The probability of a seed

7 . 9 Find the probability of the flower producing a different colour. producing a pink flower is

3. If a baby has a 0.2% chance of being born with a disability, find the probability of the baby being born without any disabilities. 4. The probability of selecting a card with the number 5 on it is 0.27. What is the probability of not selecting this card?

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5. There is a 62% chance of a student being chosen as a prefect. What is the probability of a student not being selected as a prefect?

12. The probabilities of a certain number of seeds germinating when 4 seeds are planted are as follows: P ] 4 seeds g =

6. A machine has a 1.5% chance of breaking down at any given time. What is the probability of the machine not breaking down? 7. The life of a certain type of computer is about 7 years. If the probability of its needing repairs 1 in that time is , find the 23 probability that it will not need repairs. 8. A certain traffic light has a 13 probability of of being green. 18 Find the probability of the light not being green when a car comes to the traffic light. 9. A certain organism in a river has a probability of 0.79 of surviving a flood. What is the probability of its not surviving? 10. A city has an 8.1% chance of being hit by an earthquake. What is its chance of not having an earthquake? 11. The probabilities when 3 coins are tossed are as follows: 1 8 3 P (2 heads and 1 tail) = 8 3 P (1 head and 2 tails) = 8 1 P (3 tails) = 8 P (3 heads) =



Find the probability of tossing at least one head.

P ] 3 seeds g =

P ] 2 seeds g = P ] 1 seed g =

P (0 seeds) =

4 49 8 49 16 49 18 49 3 49

Find the probability of at least 1 seed germinating.

13. The probabilities of 4 friends being chosen for a soccer team are as follows: P ] 4 chosen g =

1 15 4 P ] 3 chosen g = 15 6 P ] 2 chosen g = 15 2 P ] 1 chosen g = 15

Find the probability of (a) none of the friends being chosen (b) at least 1 of the friends being chosen.

Chapter 11 Probability

14. A dog breeder tries to produce a dog with a curly tail. If 2 puppies are born, the probabilities are as follows:

15. The probabilities of 3 new cars passing a quality control check are as follows: P ^ 3 passing h =

1 16 7 P ^ 2 passing h = 16 3 P ^ 1 passing h = 16 5 P ^ 0 passing h = 16

P ^ no curly tails h =

4 11 5 P ^ 1 curly tail h = 11 2 P ^ 2 curly tails h = 11



Find the probability that at least 1 puppy will have a curly tail.

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Find the probability that at least 1 car will fail the check.

Non-mutually exclusive events All the examples of probability given so far in this chapter are mutually exclusive. This means that if one event occurs, then another one cannot. For example, if a die is thrown, a 6 cannot occur at the same time as a 2. Sometimes, there is an overlap where more than one event can occur at the same time. We call these non-mutually exclusive events. It is important to count the possible outcomes carefully when this happens. If there are not too many outcomes, we can simply list them, but if this is difficult, we can use a Venn diagram to help.

Examples 1. One card is drawn from a set of cards numbered 1 to 10. Find the probability of drawing out an odd number or a multiple of 3.

Solution The odd cards are 1, 3, 5, 7 and 9. The multiples of 3 are 3, 6 and 9. The numbers 3 and 9 are both odd and multiples of 3. So there are 6 numbers that are odd or multiples of 3: 1, 3, 5, 6, 7 and 9 6 P (odd or multiple of 3) = 10 3 = 5 continued

The trick with this question is not to count the 3 and the 9 twice.

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Using a Venn diagram: Count all the numbers inside the two circles. There are 6 numbers inside the circles and 10 numbers altogether.

From the Venn diagram: 3 6 = . Probability is 10 5 2. In year 7 at Mt Random High School, every student must do art or music. In a group of 100 students surveyed, 47 do music and 59 do art. If one student is chosen at random from year 7, find the probability that this student does (a) both art and music (b) only art (c) only music.

Solution 47 + 59 = 106 But there are only 100 students! This means 6 students have been counted twice. i.e. 6 students do both art and music. Students doing music only: 47 − 6 = 41 Students doing art only: 59 − 6 = 53 A Venn diagram shows this information.

(a) P ] both g =

3 6 = 100 50 53 (b) P ^ art only h = 100 41 (c) P ^ music only h = 100

DID YOU KNOW? Venn diagrams are named after John Venn (1834–1923), an English probabilist and logician.

Chapter 11 Probability

There is a formula that can be used for non-mutually exclusive events. P ] A or B g = P ] A g + P ] B g − P ] A and B g Notice that P ] A g + P ] B g counts P ] A and B g twice, since it occurs in both P(A) and P(B). This can be adjusted by subtracting P(A and B).

Example From 100 cards, numbered from 1 to 100, one is selected at random. Find the probability that the card selected is even or less than 20.

Solution Some cards are both even and less than 20 (i.e. 2, 4, 6, 8, 10, 12, 14, 16, 18). 9 P (even and < 20) = 100 50 P (even) = 100 19 P (< 20) = 100 P (even or < 20) = P (even) + P (< 20) − P (even and < 20) 50 19 9 = + − 100 100 100 60 = 100 3 = 5

11.3 Exercises 1. A number is chosen at random from the numbers 1 to 20. Find the probability that the number chosen will be (a) divisible by 3 (b) less than 10 or divisible by 3 (c) a composite number (d) a composite number or a number greater than 12.

2. A set of 50 cards is labelled from 1 to 50. One card is drawn out at random. Find the probability that the card will be (a) a multiple of 5 (b) an odd number (c) a multiple of 5 or an odd number (d) a number greater than 40 or an even number (e) less than 20.

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3. A set of 26 cards, each with a different letter of the alphabet on it, is placed in a box and one is drawn out at random. Find the probability that the letter on the card drawn will be (a) a vowel (b) a vowel or one of the letters in the word ‘random’ (c) a consonant or one of the letters in the word ‘movies’. 4. A set of discs is numbered 1 to 100 and one is chosen at random. Find the probability that the number on the disc will be (a) less than 30 (b) an odd number or a number greater than 70 (c) divisible by 5 or less than 20. 5. In Lotto, a machine holds 45 balls, each with a number between 1 and 45 on it. The machine draws out one ball at a time at random. Find the probability that the first ball drawn out will be (a) less than 10 or an even number. (b) between 1 and 15 inclusive, or divisible by 6 (c) greater than 30 or an odd number. 6. A class of 28 students puts on a concert with all class members performing. If 15 dance and 19 sing in the performance, find the probability that any one student chosen at random from the class will (a) both sing and dance (b) only sing (c) only dance.

7. A survey of 80 people with dark hair or brown eyes showed that 63 had dark hair and 59 had brown eyes. If one of the people surveyed is chosen at random, find the probability that the person will (a) have dark hair but not brown eyes (b) have brown eyes but not dark hair (c) have both brown eyes and dark hair.

8. A list is made up of people with experience of either computers or digital cameras. On the list of 20 people, 13 have computer experience while 9 have experience with a digital camera. If one name is chosen at random from the list, find the probability that the person will have experience with (a) both computers and digital cameras (b) computers only (c) digital cameras only.

Chapter 11 Probability

9. In a group of 75 students, all do either History or Geography. Altogether 54 do History and 31 do Geography. If I select one student at random, find the probability that he/she will do (a) only Geography (b) both History and Geography (c) History but not Geography.

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10. In a group of 20 dogs at obedience school, 14 dogs will walk to heel and 12 will stay when told. If one dog is chosen at random, find the probability that the dog will (a) both walk to heel and stay (b) walk to heel but not stay (c) stay but not walk to heel.

Multi-Stage Events Product rule of probability

Class Discussion Break up into pairs and try these experiments with one doing the activity and one recording the results. 1. Toss two coins as many times as you can in a 5 minute period and record the results in the table: Result

Two heads

One head and one tail

Two tails

Tally Compare your results with others in the class. What do you notice? Is this surprising? 2. Roll two dice as many times as you can in a 5 minute period, find the total of the two uppermost numbers on the dice and record the results in the table: Total

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Tally Compare your results with others in the class. What do you notice? Is this surprising? Why don’t these results appear to be equally likely?

The counting of all possible outcomes (the sample space) is important. This is why we use tables and tree diagrams.

You learned about tree diagrams in your earlier studies.

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Examples Compare these probabilities with your results in the experiments.

Find the sample space and the probability of each outcome for each question by using a table or tree diagram. 1. Tossing two coins

Solution Using a table gives H

T

H

HH

HT

T

TH

TT

Using a tree diagram gives H H T H T T

Since there are four separate outcomes (HH, HT, TH, TT) each outcome 1 has a probability of . 4 1 Remember that each outcome when tossing 1 coin is . 2 1 1 1 Notice that ´ = . 2 2 4 2. Rolling 2 dice and recording the sum of the uppermost numbers. Can you see why a tree diagram is too difficult here?

Solution A tree diagram would be too difficult to draw for this question. Using a table: 1

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1 36 1 Remember that each outcome when rolling 1 die is . 6 1 1 1 Notice that ´ = 6 6 36

Since there are 36 outcomes, each has a probability of

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If A and B are independent events, then the probability of both occurring is the product of their probabilities. P ] AB g = P ] A g $ P ] B g

Examples 1 . Find the 6 probability of getting a double 6 when rolling two dice. 1. The probability of getting a 6 when rolling a die is

Solution P ] double 6 g =

1 1 ´ 6 6 1 = 36

7 2. The probability that a certain missile will hit a target is . Find the 8 probability that the missile will (a) hit two targets (b) miss two targets.

Solution 7 7 (a) P ] 2 hits g = ´ 8 8 49 = 64 7 (b) P ] miss g = 1 − 8 1 = 8 1 1 P ] 2 misses g = ´ 8 8 1 = 64

Sometimes the outcomes change when looking at more than one event.

Using these answers, could you calculate the probability that the missile hits one target and not the other?

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Examples 1. Maryam buys 5 tickets in a raffle in which 95 tickets are sold altogether. There are two prizes in the raffle. What is the probability of her (a) winning both first and second prizes? (b) winning neither prize? (c) winning at least one of the prizes?

Solution (a) Probability of winning first prize =

5 95

After winning first prize, Maryam’s winning ticket is taken out of the draw. She then has 4 tickets left in the raffle out of a total of 94 tickets left. 4 Probability of winning second prize = 94 5 4 P ] WW g = ´ 95 94 2 = 893 5 (b) Probability of not winning first prize = 1 − 95 90 = 95 After not winning first prize, Maryam’s 5 tickets are all left in the draw, but the winning ticket is taken out, leaving 94 tickets in the raffle. 5 Probability of winning second prize = 94 5 Probability of not winning second prize = 1 − 94 89 = 94

P ] LL g =

90 89 ´ 95 94 801 = 893

(c) P ] at least one W g = 1 − P ] LL g 801 =1− 893 92 = 893 2. I choose 3 balls at random from a bag containing 7 blue and 5 red balls. (a) Find the probability of getting 3 blue balls if (i) I replace each ball before choosing the next one (ii) I don’t replace each ball before choosing the next one. (b) Find the probability of getting at least one red ball (without replacement).

Chapter 11 Probability

Solution (a) (i)

P ]Bg =

7 12

So P ] BBB g = 7 ´ 7 ´ 7 12 12 12 343 = 1728 7 (ii) P ] B g = 12 After the first blue ball has been chosen, the bag now contains 6 blue and 5 red balls. P ] 2nd B g =

6 11 After the second blue ball has been chosen, the bag contains 5 blue and 5 red balls. P ] 3rd B g =

5 10 6 5 7 So P ] BBB g = ´ ´ 12 11 10 7 = 44 (b) P ] at least one R g = 1 − ] no R g = 1 − P ] BBB g 7 =1− 44 37 = 44

11.4 Exercises 1. If 2 dice are thrown, find the probability of throwing two 6’s. 2. Find the probability of getting 2 heads if a coin is tossed twice. 3. A coin is tossed 3 times. Find the probability of tossing 3 tails. 4. A card has a picture on one side and is blank on the other. If the card is thrown into the air twice, find the probability that it will land with the picture side up both times.

5. A box contains 2 black balls, 5 red balls and 4 green balls. If I draw out 2 balls at random, replacing the first before drawing out the second, find the probability that they will both be red. 6. The probability of a conveyor belt in a factory breaking down at any one time is 0.21. If the factory has 2 conveyor belts, find the probability that at any one time (a) both machines will break down (b) neither machine will break down.

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7. The probability of a certain plant flowering is 93%. If a nursery has 3 of these plants, find the probability that they will all flower. 8. An archery student has a 69% chance of hitting a target. If she fires 3 arrows at a target, find the probability that she will hit the target each time. 9. A bag contains 8 yellow and 6 green lollies. If I choose 2 lollies at random, find the probability that they will both be green (a) if I replace the first lolly before selecting the second (b) if I don’t replace the first lolly. 10. I buy 10 tickets in a raffle in which 250 tickets are sold. Find the probability of winning both first and second prizes. 11. Two cards are drawn from a deck of 20 red and 25 blue cards (without replacement). Find the probability that they will both be red. 12. Find the probability of winning the first 3 prizes in a raffle if Peter buys 5 tickets and 100 tickets are sold altogether. 13. The probability of a pair of small parrots breeding an albino bird 2 is . If they lay three eggs, find 33 the probability of the pair (a) not breeding any albinos (b) having all three albinos (c) breeding at least one albino.

14. A photocopier has a paper jam on average around once every 2400 sheets of paper. (a) What is the probability that a particular sheet of paper will jam? (b) What is the probability that two particular sheets of paper will jam? (c) What is the probability that two particular sheets of paper will not jam?

15. In Yahtzee, 5 dice are rolled. Find the probability of rolling (a) five 6’s (b) no 6’s (c) at least one 6. 16. The probability of a faulty computer part being manufactured at Omega 3 Computer Factory is . 5000 If 2 computer parts are examined, find the probability that (a) both are faulty (b) neither are faulty (c) at least one is faulty. 17. A set of 10 cards is numbered 1 to 10 and two are drawn out at random with replacement. Find the probability of drawing (a) two odd numbers (b) two numbers that are divisible by 3 (c) two numbers less than 4.

Chapter 11 Probability

18. A bag contains 5 white, 4 black and 3 red marbles. If 2 marbles are selected from the bag at random without replacement, find the probability of selecting (a) two red marbles (b) two black marbles (c) no white marbles (d) at least one white marble. 19. The probability of an arrow hitting a target is 85%. If 3 arrows are shot, find the probability as a percentage, correct to 2 decimal places, of

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(a) all arrows hitting the target (b) no arrows hitting the target (c) at least one arrow hitting the target. 20. A coin is tossed n times. Find the probability in terms of n of tossing (a) all heads (b) no tails (c) at least one tail.

Tree diagrams and probability trees When using the product rule to find the probability of successive events occurring, sometimes there is more than one possible result. For example, when tossing two coins, there are two ways of getting a head and a tail (HT and TH). We add these results together. P ] A or B g = P ] A g + P ] B g

This is called the addition rule of probability.

We use tree diagrams or probability trees to combine the product and addition rules. We use the product rule by multiplying along the branches and the addition rule by adding up the probabilities from different branches.

Examples 1. If 2 coins are tossed, find the probability of tossing a head and a tail.

Solution

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P ] head and tail g = P (HT ) + P (TH ) 1 1 1 1 = c ´ m+c ´ m 2 2 2 2 1 1 = + 4 4 1 = 2 2. A person has probability of 0.2 of winning a prize in a competition. If he enters 3 competitions, find the probability of his winning (a) 2 competitions (b) at least 1 competition.

Solution

Probability of losing is 1 – 0.2.

(a) Probability of losing is 0.8 P (2W) = P (WWL) + P (WLW) + P (LWW ) = (0.2 ´ 0.2 ´ 0.8) + (0.2 ´ 0.8 ´ 0.2) + (0.8 ´ 0.2 ´ 0.2) = 0.032 + 0.032 + 0.032 = 0.096 (b) P (at least one W ) = 1 − P (LLL) = 1 − (0.8 ´ 0.8 ´ 0.8) = 1 − 0.512 = 0.488

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3. A bag contains 3 red, 4 white and 7 blue marbles. Two marbles are drawn at random from the bag (a) replacing the first before the second is drawn (b) without replacement Find the probability of drawing out a red and a white marble in these cases.

Solution (a)

P (R and W ) = P (RW ) + P (WR) 3 3 4 4 =c ´ ´ m+c m 14 14 14 14 12 12 = + 196 196 6 = 49 (b)

The probabilities in each set of branches must add up to 1.

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P (R and W ) = P (RW ) + P (WR) 3 3 4 4 ´ ´ =c m+c m 14 13 14 13 12 12 = + 182 182 12 = 91

11.5 Exercises 1. Two coins are tossed. Find the probability of getting (a) 2 heads (b) a head followed by a tail (c) a head and a tail. 2. Three coins are tossed. Find the probability of getting (a) 3 tails (b) 2 heads and 1 tail (c) at least 1 head. 3. In a set of 30 cards, each one has a number on it from 1 to 30. If 1 card is drawn out, then replaced and another drawn out, find the probability of getting (a) two 8s (b) a 3 on the first card and an 18 on the second card (c) a 3 on one card and an 18 on the other card. 4. Five cards are labelled A, B, C, D and E. If 2 are selected at random, with replacement, find the probability that they will be (a) both As (b) an A and a D. 5. A bag contains 5 red marbles and 8 blue marbles. If 2 marbles are chosen at random, with the first

replaced before the second is drawn out, find the probability of getting (a) 2 red marbles (b) a red and a blue marble. 6. A certain breed of cat has a 35% probability of producing a white kitten. If a cat has 3 kittens, find the probability that she will produce (a) no white kittens (b) 2 white kittens (c) at least 1 white kitten. 7. The probability of a certain type of photocopier in a school needing a service on any one day is 0.3. Find the probability that a school with 2 of these photocopiers will need to service, on a particular day, (a) 1 machine (b) both machines (c) neither of them 8. The probability of rain in May 3 each year is given by .A 10 school holds a fete in May for three years running. Find the probability that it will rain at (a) 2 of the fetes (b) 1 fete (c) at least 1 fete

Chapter 11 Probability

9. A certain type of plant has a probability of 0.85 of producing a variegated leaf. If I grow 3 of these plants, find the probability of getting a variegated leaf in (a) 2 of the plants (b) none of the plants (c) at least 1 plant.

13. Mary buys 20 tickets in a lottery that has 5000 tickets altogether. Find the probability that Mary will win (a) first and second prize (b) second prize only (c) neither first nor second prize. 14. Two musicians are selected at random to lead their band. One person is chosen from Band A, which has 8 females and 7 males, and the other is chosen from Band B, which has 6 females and 9 males. Find the probability of choosing (a) 2 females (b) 1 female and 1 male.

10. A bag contains 6 white balls and 5 green balls. If 2 balls are chosen at random, find the probability of getting a white and a green ball (a) with replacement (b) without replacement. 11. A bag contains 3 yellow balls, 4 pink balls and 2 black balls. If 2 balls are chosen at random, find the probability of getting a yellow and a black ball (a) with replacement (b) without replacement. 12. Alan buys 4 tickets in a raffle in which 100 tickets are sold altogether. There are two prizes in the raffle. Find the probability that Alan will win (a) first prize (b) both prizes (c) 1 prize (d) no prizes (e) at least 1 prize.

15. The two machines in a workshop 1 each have a probability of 45 of breaking down. Find the probability that at any one time (a) neither machine will be broken down (b) 1 machine will be broken down. 16. Two tennis players are said to 2 have a probability of and 5 3 respectively of winning a 4 tournament. Find the probability that (a) 1 of them will win (b) neither one will win. 17. If 4 dice are thrown, find the probability that the dice will have (a) four 6’s (b) only one 6 (c) at least one 6.

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18. In a batch of 100 cars, past experience would suggest that 3 could be faulty. If 3 cars are selected at random, find the probability that (a) 1 is faulty (b) none are faulty (c) all 3 cars are faulty. 19. In a certain poll, 46% of people surveyed liked the current government, 42% liked the opposition and 12% had no preference. If 2 people from the survey are selected at random, find the probability that (a) both will prefer the opposition (b) 1 will prefer the government and the other will have no preference (c) both will prefer the government. 20. A manufacturer of X brand of soft drink surveyed a city and found that 31 people liked X drinks best, 19 liked another brand better and 5 did not drink soft drink. If any 2 people are selected at random from that city, find the probability that (a) one person would like the X brand of soft drink (b) both people would not drink soft drink. 21. A bag contains 5 red, 6 blue, 2 white and 7 green balls. If 2 are selected at random (without replacement), find the probability of getting (a) 2 red balls (b) a blue and a white ball (c) 2 green balls (d) at least 1 red ball.

22. In a group of people, 32 are Australian born, 12 were born in Asia and 7 were born in Europe. If 2 of the people are selected at random, find the probability that (a) they were both born in Asia (b) at least 1 of them will be Australian born (c) both were born in Europe. 23. There are 34 men and 32 women at a party. Of these, 13 men and 19 women are married. If 2 people are chosen at random, find the probability that (a) both will be men (b) 1 will be a married woman and the other an unmarried man (c) both will be married. 24. In a certain city, the probability that the pollution level will be high is 0.27. If the pollution is monitored for 4 successive days, find the probability that the pollution levels will be (a) high on 2 days (b) high on 1 day (c) low on at least 1 day.

Chapter 11 Probability

25. At City Heights School it was found that 75% of students in year 12 study 13 units, 21% study 12 units and 4% study 11 units. If 2 students are selected at random from year 12, find the probability that (a) 1 student will study 12 units (b) at least 1 student will study 13 units. 26. Three dice are rolled. Find the probability of rolling (a) 3 sixes (b) 2 sixes (c) at least 1 six. 27. A set of 5 cards, each labelled with the letters A, B, C, D and E, is placed in a hat and two selected at random without replacement. Find the probability of getting (a) D and E (b) Neither D nor E on either card (c) At least one D.

28. The ratio of girls to boys at a school is four to five. Two students are surveyed at random from the school. Find the probability that the students are (a) both boys (b) a girl and a boy (c) at least one girl. 29. The number of cats to dogs at a pet hotel is in the ratio of 4 to 7. If 3 pets are chosen at random, find the probability that (a) they are all dogs (b) just one is a dog (c) at least one is a cat. 30. A set of 20 cards is numbered 1 to 20 and three are selected at random with replacement. Find the probability of selecting (a) all three 10’s (b) no 10’s (c) at least one 10.

Counting techniques As you can see from the previous section, the hardest part of calculating probabilities is finding all the possible outcomes. You have used lists, tables, tree diagrams and probability trees to help find the sample space. However, these can become quite difficult to use in some cases. Think of how you would list all the possible outcomes when rolling 3 dice! There are many examples where counting techniques are useful. For example, in the early days, phone numbers used to have fewer digits, but these ran out when too many people started to have a telephone. Now mobile phone numbers have 10 digits, but in the future as more and more people use them, we might need to add more digits. In some areas of science, for example, in genetics, counting of molecules on strands of DNA can be challenging. You studied permutations and combinations in the Preliminary course. We will revise this work here.

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Fundamental counting principle The product rule uses this.

If 1 event can happen in p different ways and after this another event can happen in q different ways, then the 2 successive events can happen in pq different ways.

Examples 1. The numberplate on a car has 2 letters, followed by 4 numbers. How many different numberplates of this type are possible?

Solution There are 26 letters and 10 numbers (0 to 9) possible for each position in the numberplate. Total number = 26 ´ 26 ´ 10 ´ 10 ´ 10 ´ 10 = 26 2 ´ 10 4 = 6 760 000 So 6 760 000 numberplates are possible. 2. I have 12 pairs of earrings, 3 necklaces, 8 rings and 2 watches in my jewellery box. (a) If I can wear any combination of earrings, necklaces, rings and watches, how many different sets of jewellery can I wear? (b) If my friend makes a guess at the combination of jewellery that I will wear, what is the probability that she will guess correctly?

Solution (a) Total number = 12 ´ 3 ´ 8 ´ 2 = 576 1 (b) P ^ correct guess h = 576

11.6 Exercises 1. A combination lock has 4 dials, each with 10 digits. How many combinations are possible?

2. A certain type of serial number on a television set is made up of 5 numbers, followed by a letter. How many serial numbers of this type are available?

Chapter 11 Probability

3. A personalised car numberplate can fit up to 6 letters on it. If I could use any letter of the alphabet, or a space, in any position, how many different combinations could be formed? 4. A personal identification number (PIN) is made up of 4 digits. How many different PINs are possible? 5. Jan saw a car leaving the bank after a robbery. She remembered all of its numberplate except the last number. How many cars could there be with this numberplate? 6. A poker machine has 5 reels, each with 12 symbols on it. How many different combinations are possible? 7. A local library has 59 books on drama, 102 books on ballet and 87 books on gymnastics. I ask my friend to borrow a particular book on each of these subjects, but he loses my list. If he chooses 1 of each type of book at random, find the probability that he will choose the 3 books that I wanted. 8. A company that manufactures radios labels each radio with a serial number made up of 2 numbers and 3 letters. How many radios can have this type of serial number? 9. To win a trifecta in a race, a person has to pick the horses that come first, second and third in the race. For a certain race, Marie wishes to bet on

every combination possible to win the trifecta. If there are 12 horses in the race, how many combinations will she bet on? 10. A security door unlocks when a person presses a certain combination of 6 numbers. If I try a particular combination at random, find the probability that I will be able to unlock the door. 11. The NSW and ACT postcodes all have 4 digits and start with 2. How many different postcodes are possible? 12. Many telephone numbers in Sydney have 8 digits. If the first digit is not allowed to be zero, how many telephone numbers are possible? 13. The game of Yahtzee involves tossing 5 dice. (a) How many ways can the 5 dice land? (b) What is the probability of tossing 5 sixes? 14. Wendy orders a 3 course dinner at a restaurant that offers 8 entrées, 5 main courses and 7 desserts. What is the probability that she has ordered the same combination as her friend? 15. A certain brand of car has a choice of 3 engine sizes and 7 colours, a choice of manual or automatic, and a choice of a sunroof or air conditioning. How many different combinations are possible?

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Permutations A permutation describes an arrangement of objects in a certain order.

Example A number from 1 to 20 is written on each of 20 cards. If 3 cards are chosen randomly, without replacement, find the possible number of ways the cards can be chosen.

Solution The first card can be any of the 20 numbers. The second card can be any of the remaining 19 numbers. The third card can be any of the remaining 18 numbers. ∴ the number of ways the cards can be chosen is 20 ´ 19 ´ 18 = 6840

How many ways could 3 cards be chosen with replacement?

Permutation n Pr is the number of ways of making ordered selections of r objects from a total of n objects. The number of ways to choose r objects in order from a total of n objects is given by n ] n − 1 g ] n − 2 g f ] n − r + 1 g .

n

You studied permutations in the Preliminary course.

Pr =

n! ]n − r g!

Proof n

Pr = n (n − 1) (n − 2) . . . (n − r + 1) = n (n − 1) (n − 2) . . . (n − r + 1) ´ n ] n − 1 g ] n − 2 g . . . 3. 2. 1 ] n − r g ] n − r − 1 g . . . 3. 2. 1 n! = ]n − r g!

] n − r g ] n − r − 1 g . . . 3. 2. 1 ] n − r g ] n − r − 1 g . . . 3. 2. 1

=

n

Proof n! ]n − r g! n! ` n Pn = ]n − ng! n! = 0! n! = 1 = n! n

Remember that 0! = 1.

Pr =

Pn = n!

Chapter 11 Probability

Examples 1. (a) Find the number of 4 digit numbers that can be made using the numbers 0 to 9 if each number can only be used once. (b) How many 4 digit numbers greater than 6000 can be formed?

Solution There are 10 digits from 0 to 9. (a) The 1st digit can be any of the 10 numbers. The 2nd digit can be any of the remaining 9 numbers. The 3rd digit can be any of the remaining 8 numbers. The 4th digit can be any of the remaining 7 numbers.



Total permutations = 10 ´ 9 ´ 8 ´ 7 = 5040 10 10! P4 = or ] 10 − 4 g ! 10! = 6! = 5040

(b) The 1st digit can be 6, 7, 8 or 9 (4 possible digits). The 2nd digit can be any of the remaining 9 numbers. The 3rd digit can be any of the remaining 8 numbers. The 4th digit can be any of the remaining 7 numbers.

Total arrangements = 4 ´ 9 ´ 8 ´ 7 = 2016

or There are 4 ways to get the 1st digit. The possible arrangements for the remaining 3 digits are 9 P3 . Total arrangements = 4 ´ 9 P3 = 4 ´ 504 = 2016

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2. (a) In how many ways can 9 people sit around a circular table with no conditions restricting where anyone sits? (b) If 2 people want to sit together, what is the probability that they will be, if they are seated at random?

Solution

If A can sit anywhere, we fix this position and arrange the others.

(a) The 1st person can sit anywhere. The 2nd person can sit in any of the remaining 8 seats. The 3rd person can sit in any of 7 seats, and so on. ∴ total arrangements depend on the seating of 8 people Total arrangements = 8! = 40 320

(b)

The 2 people can sit together in 2! ways (AB or BA). The remaining 7 people can sit in 7! ways. Total arrangements = 2! ´ 7! = 10 080 2 ´ 7! P ^ 2 sitting together h = 8! 10 080 = 40 320 1 = 4



3. (a) How many ways can the letters of the word ‘PROBABLE’ be arranged? (b) If I jumble the letters in the word ‘PROBABLE’ and make up a word at random, find the probability that the word I choose will be the reverse of PROBABLE (i.e. ELBABORP).

Chapter 11 Probability

Solution (a) PROBABLE has 8 letters, with 2 B’s. If each B were different, i.e. B1 and B2, then there would be 8! arrangements. However, we cannot tell the difference between the 2 B’s. Since there are 2! ways to arrange the 2 B’s, there are 2! arrangements of the word ‘PROBABLE’ that look the same. 8! ` total arrangements = 2! = 20 160 1 (b) P ] ELBABORP g = 20 160

11.7 Exercises 1. Evaluate, then check answers on the calculator. (a) 6 P4 (b) 7 P6 (c) 9 P1 (d) 5 P3 (e) 8 P6 2. If I have 10 cards, each labelled with a different number from 1 to 10, find how many numbers are possible if selecting (without replacement) (a) 2 cards (b) 6 cards (c) 5 cards (d) 3 cards (e) 8 cards. 3. A 3 digit number is to be made from the digits 2, 3, 4, 5, 6 and 7. (a) How many numbers can be made if no digit may be used more than once in the same number? (b) In how many ways can an even number be made from these digits? (c) How many numbers over 600 can be made?

4. A 4 digit number is to be made from the digits 1 to 9, with no digit allowed more than once in the same number. (a) How many numbers can be made? (b) In how many ways can an odd number be made? (c) How many numbers less than 3000 can be made? 5. (a) How many 1, 2, 3 or 4 digit numbers can be made using the digits 1 to 4, with no digit allowed more than once? (b) How many numbers less than 100 can be made? 6. How many 3 letter combinations can be made from the word SWITZERLAND? 7. How many arrangements of the digits 1, 2, 3 and 4 can be made without using any digit more than once in the same number? 8. How many 3 digit numbers can be made from the digits 1, 2, 3, 4 and 5 without using any digit twice in the same number?

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9. In how many ways can a group of 7 people be arranged (a) in a straight line (b) around a circle? 10. In how many ways can a group of 4 people be arranged (a) in a straight line (b) around a circle?

A bracelet can be turned over so it looks the same either side.

11. A queue has 4 boys and 4 girls standing in line. Find how many different arrangements of the line are possible if (a) the boys and girls can stand anywhere in the line (b) the boys and girls alternate (c) 2 particular girls wish to stand together (d) all the boys stand together. (e) Also find the probability that 3 particular people will be in the queue together if the queue forms randomly. 12. A table has 4 boys and 4 girls sitting around it. (a) Find the number of ways of sitting possible if the boys and girls can sit anywhere around the table. (b) If the seating is arranged at random, find the probability that (i) 2 particular girls will sit together, and (ii) all the boys will sit together. 13. How many ways can 10 people be arranged in a (a) line? (b) circle?

14. At a dinner party 6 people sit around a table. (a) How many different arrangements are possible? (b) Find the probability that if seating is at random, 2 friends will sit apart. (c) How many ways can the seating be arranged so that 3 particular people will sit together? 15. (a)  Nine beads are arranged randomly in a line. How many arrangements are possible? (b) If the beads are placed in a circle, how many ways can this be done? (c) The beads are arranged on a bracelet. How many different ways are possible? 16. Find the probability that in a circle of 20 people, 2 particular people will be together. 17. How many different arrangements can be made from the following words? Note that the arrangements need not be proper words. (a) MATHS (b) WORD (c) ELEPHANT (d) POPULAR (e) SAUSAGE (f) TEPEE (g) BLACKBOARD (h) PERCENTAGE (i) ENGINEERING (j) SUPERMARKET

Chapter 11 Probability

18. Find out how many arrangements are possible from the word STUDIO if (a) any grouping of letters is allowed (b) the T and D are together (c) the vowels are together (d) the vowels and consonants alternate (e) the letter S is not the first letter (f) the letter O is not first or last.

19. In how many ways can 10 boys be arranged in a line if (a) the first boy in the line is always the same (b) the first boy and the last boy in the line are always the same? 20. Twelve differently coloured beads are arranged around a necklace. How many different arrangements are possible?

Combinations The permutation n Pr is the number of arrangements possible for an ordered selection of r objects from n objects. That is, the selection AB and BA are different, as they are in a different order. When the order is not important, that is, when AB and BA mean the same thing, the number of arrangements is called a combination. You studied combinations in the previous chapter. The number of ways of making unordered selections of r objects from n objects is given by n C r .

Proof Let n Pr be the ordered selection of r objects from n objects. There are r! ways of arranging the r objects. n P If order is unimportant, then the unordered selection of r objects is given by  r . r!

n! ]n − r g! = r! r! n! 1 ´ = ] n − r g ! r! n! = ] n − r g !r! = nCr

n

Pr

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Examples 1. (a) A committee of 4 people is formed from a group of 11 people. In how many different ways can the committee be formed? (b) If the group consists of 5 men and 6 women, how many ways can the committee be formed with 1 man and 3 women?

Solution (a) The order of the committee is not important (e.g. a committee of David and Mary is the same as a committee of Mary and David). Number of committees = 11 C 4 = 330 (b) One man can be chosen from 5 in 5 C 1 or 5 ways. Three women can be chosen from 6 in 6 C 3 or 20 ways. Total number of arrangements = 5 C 1 ´ 6 C 3 = 5 ´ 20 = 100



2. (a) Twelve people apply for a scholarship to boarding school, but only 5 scholarships are available. How many different combinations of people are possible for the 5 scholarships? (b) Out of the 12 people, 4 are from Sydney and 8 are from the country. If 3 scholarships are awarded to country people, how many combinations are possible?

Solution (a) 12 C 5 = 792 (b) The 4 Sydney people can get 2 scholarships in 4 C 2 ways. The 8 country people can get 3 scholarships in 8 C 3 ways. Total combinations = 4 C2 ´ 8 C3 = 6 ´ 56 = 336 3. A team of 4 men and 5 women is to be chosen at random from a group of 8 male and 7 female swimmers. If Craig and Tracey are both hoping to be chosen, find the probability that (a) both will be chosen (b) neither will be chosen.

Chapter 11 Probability

Solution The number of possible teams = 8 C4 ´ 7 C5 = 1470 (a) If Craig is chosen, then 3 of the other 7 men need to be chosen, i.e. 7 C 3 . If Tracey is chosen, then 4 of the other 6 women need to be chosen, i.e. 6 C 4 .

Number of combinations = 7 C 3 ´ 6 C 4 = 525 525 5 Probability = = 1470 14 (b) If Craig and Tracey are not included in the team, then 4 men out of the other 7 are chosen, and 5 women out of the other 6 are chosen.



Number of combinations = 7 C 4 ´ 6 C 5 = 210 210 1 Probability = = 1470 7

11.8 Exercises 1. In a class of 20 students, 2 are selected as class prefects. If the selection is made at random, in how many ways is the selection possible? 2. A group of 8 tennis players all have an equal chance of being chosen in a team. If the team can only have 5 players, in how many ways can the team be selected? 3. In a group of 6 apprentices, a team of 4 is put on a special job. If the selection is made at random, in how many different ways can the selection of the team be made? 4. A debating team of 4 is chosen from a class of 15. How many different combinations are possible (a) if there are no restrictions on who is in the team

(b) if 1 particular person is to be included in the team (c) if 2 particular people are to be included? 5. A committee of 3 women and 5 men is to be selected at random from 7 women and 7 men. (a) Find the number of ways that the committee can be chosen. (b) Among the group is a married couple. Find the probability that they will both be selected. 6. Out of 25 students who study drama, 3 are chosen to be in a play. (a) In how many ways can this be done if the selection is random? (b) A brother and sister both do drama. Find the probability that neither of them will be selected for the play.

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7. There are 6 books about ancient Rome on a certain shelf in the library. If there are 20 books on the shelf altogether and I choose 6 at random, find the probability that they will all be about ancient Rome. 8. An excursion is arranged for a class of 31 students. However, there is only room for 20 students in the bus. If 20 students are selected at random from the class, in how many ways can this be done? 9. To win Lotto, you choose 6 numbers out of a total of 45. (a) In how many ways can this be done? (b) What is the probability of winning first prize in Lotto if you play 4 games? (c) Find the probability of winning Lotto if you play 100 games. 10. To play the Australian Soccer Pools you choose 6 numbers out of 38. What is the probability of winning if you play 10 games? 11. A committee of 3 people is formed from a group of 9 people. (a) In how many different ways can the committee be formed? (b) If the group consists of 6 men and 3 women, how many ways can the committee be formed with 2 men and 1 woman? 12. A team of 3 girls and 3 boys is chosen from a group of 15 girls and 12 boys to represent their school. In how many different ways can the team be formed?

13. There are 9 people who are applying for 4 jobs. (a) If the jobs are allocated randomly, how many different combinations of people are possible for the 4 jobs? (b) Out of the 9 people, 5 have their HSC. If 3 jobs require the HSC, how many ways can the 4 jobs be allocated? 14. Thirty people apply for a special housing package, but only 12 packages are available, and are allocated randomly. (a) What is the probability that a particular person applying will get a house? (b) Out of the 12 packages, 5 are in Sydney and 7 are in the Blue Mountains. If 19 people apply for the Sydney houses and the rest apply for the Blue Mountains, in how many ways can the houses be allocated? 15. Of a group of 16 children at preschool, only 5 are allowed on the climbing equipment at one time. (a) How many different combinations of children are possible? (b) Out of the 16 children, 7 are under 3 years old. If 2 children under 3 years and 3 over 3 years are allowed on the climbing equipment, how many ways can this be done? (c) Find the probability that Allan, who is 2 years old, and his friend Hannah, who is 4 years old, will both be included in the group chosen to play on the climbing equipment.

Chapter 11 Probability

16. A school committee is to be made up of 5 teachers, 4 students and 3 parents. (a) If 12 teachers, 25 students and 7 parents apply to be on the committee, which is chosen at random, how many possible committees could be formed? (b) If Jan and her mother both apply, find the probability that both will be chosen for the committee. 17. Tom chooses 2 kittens at random from a litter of 3 male and 2 female kittens. (a) How many different selections could he make if his selection is random? (b) If Tom selects 1 male and 1 female kitten, how many ways can he do this? 18. A sample of 3 coins is taken at random from a bag containing 8 ten cent coins and 8 twenty cent coins. (a) In how many ways can the selection be made? (b) Find the probability that a particular ten cent coin will be chosen, if 2 ten cent and 1 twenty cent coins are chosen.

19. A basketball team of 2 men and 3 women is to be chosen at random from a group of 5 male and 7 female players. (a) In how many ways can the selection be made? (b) If Mike has to be in the team, in how many ways can the selection be made? (c) If George and his girlfriend both try out for the team, find the probability that neither will be chosen in the team. 20. A group of 8 people is chosen at random from a class of 30 students to go on a hike. (a) In how many ways can the group be selected? (b) Out of 12 teachers, 2 are chosen to go on the hike with the students. In how many ways can the teachers be selected at random? (c) Mr Baldwin’s favourite student is Paula. Find the probability that they will both be chosen.

Binomial probability distribution In the last chapter you saw how Pascal’s triangle can be written using combinations, or the coefficients of binomial products. Pascal’s triangle is also useful in probability.

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Class Investigation 1. By using a tree diagram or otherwise, complete the table with the number of ways of getting different combinations of heads and tails when tossing coins. Number of coins

Number of combinations

1

1 head

1 tail

2

2 heads

1 head

2 tails

1 tail 3

4 5

3 heads

4 heads 5 heads

2 heads

1 head

3 tails

1 tail

2 tails

3 heads

2 heads

1 head

1 tail

2 tails

3 tails

4 heads

3 heads

2 heads

1 head

1 tail

2 tails

3 tails

4 tails

4 tails 5 tails

2. Can you see a link between these results and Pascal’s triangle?

Let the probability of heads be p and the probability of tails be q when tossing a coin. Investigate all the possible outcomes for tossing different numbers of coins.

Example A tree diagram shows all possible outcomes for when 3 coins are tossed.

Solution Total probabilities = p 3 + 3p 2 q + 3pq 2 + q 3 = (p + q) 3

Chapter 11 Probability

If p is the probability of success and q is the probability of failure for an event, then the probability of r successes in n independent events is given by P ] r successes g = n C r p r q n − r

Proof Let p be the probability of 1 success. Then the probability of r successes is p ´ p ´ p ´ f ´ p. i.e. p r Let the probability of failure be q. Then r successes in n events means (n − r) failures. The probability of (n − r) failures is q ´ q ´ q ´ … ´ q. i.e. q n − r There are n C r ways of getting r successes from n events. ` P ] r successes g = nCr p r q n − r This means that when n trials are performed, with probabilities p and q possible, then the total of probabilities is ^ p + q hn .

Examples 1 of producing white flowers. If 3 4 plants are grown, find the probability that 1 plant will produce white flowers. 1. A certain plant has a probability of

Solution 1 3 2 P (Not white) = 3 P (W ) =

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If we draw a probability tree, we can see all the different ways in which 1 plant out of the 4 will produce white flowers.

There are 4 different ways this can happen: WNNN NWNN NNWN NNNW However, we could find the number of ways this can happen without a probability tree. We use combinations. Number of ways = 4 C 1 =4 To find the probability of each combination, we multiply along the branches of the probability tree. e.g. P ] WNNN g =

1 2 2 2 ´ ´ ´ 3 3 3 3

Each other combination has the same probability (the fractions are just in a different order). 1 1 2 3 So each combination has probability c m ´ c m . 3 3 1 3 1 2 So P (1 white) = 4 C 1 c m c m 3 3 32 = 81 2. Maria tosses 8 coins. Find the probability of tossing 5 heads.

Solution 1 2 1 P (T) = 2 P (H ) =

There are 8C5 different ways of getting 5 heads when tossing 8 coins. 5 1 3 The probability of getting 5 heads is c 1 m and 3 tails is c m . 2 2

Chapter 11 Probability

1 5 1 3 So P (5 heads) = 8 C 5 c m c m 2 2 7 = 32 3. If I throw 5 dice in a game of Yahtzee, find the probability of throwing 3 sixes.

Solution P (6) =

5 1 and P ] not 6 g = 6 6

1 3 5 5−3 P ] 3 sixes g = 5 C 3 c m c m 6 6 3 5 2 1 = 5 C3 c m c m 6 6 25 1 ´ = 10 ´ 216 36 250 = 7776 125 = 3888 4. A car assembly plant has a certain machine with an average probability of 0.1 of breaking down. If the assembly plant has 8 of these machines, what is the probability, correct to 3 decimal places, that at least 6 will be in good working order at any one time?

Solution P ] broken down g = 0.1 ` P (working) = 0.9 We want to find the probability of at least 6 working machines. This means that 6 or 7 or 8 are working. P (6) = 8 C 6 ] 0.9 g6 ] 0.1 g2 P (7) = 8 C 7 ] 0.9 g7 ] 0.1 g1 P (8) = 8 C 8 ] 0.9 g8 ] 0.1 g0

Remember when there is more than one answer, we add them together. So P (at least 6 working) = P (6) + P (7) + P (8) = 8 C 6 (0.9) 6 (0.1) 2 + 8 C 7 (0.9) 7 (0.1) 1 + 8 C 8 (0.9) 8 (0.1) 0 = 0.962

We can use the binomial theorem for further probability questions.

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Example A supermarket display contains 5 different brands of tomato paste, including TomTom brand. Over the period of a week, 80 people randomly buy tomato paste. (a) What is the greatest number of people likely to buy the TomTom (TT) brand? (b) Find the probability of this number of people buying the TT brand, correct to 2 decimal places.

Solution (a) There are 5 brands and people are buying these randomly. 1 So P ] TT g = 5 4 P ] other brands g = 5 Let k be the number of people that buy the TT brand. 1 k 4 80 − k P (TT) = 80 Ck c m c m 5 5 To find the greatest number of people, we find the value of k that gives the greatest value of P(TT) You learned how to find the greatest value of coefficients in the previous chapter.



1 k 4 80 − k Tk + 1 = 80 C k c m c m 5 5



1 k − 1 4 80 − Tk = 80 C k − 1 c m c m 5 5



Comparing coefficients: Tk + 1 Tk

1 k 4 80 − k Ck c m c m 5 5 = ] g k−1 1 4 80 − k − 1 80 Ck − 1 c m c m 5 5 1 80 Ck c m 5 = 4 80 Ck − 1 c m 5 80 Ck 1 4 ´ ¸ = 80 Ck − 1 5 5 80

80



]k − 1 g

= =

80

Ck

Ck − 1 80 Ck

´

4 80 C k − 1

1 5 ´ 5 4

Chapter 11 Probability

=

=

= = =





80! ] 80 − k g !k! 4 ] 80! g 6 80 − ] k − 1 g @ ! (k − 1)! 6 80 − ] k − 1 g @ ! (k − 1)! 80! ´ ] 80 − k g !k! 4 ] 80! g 80 − ] k − 1 g 4k 80 − k + 1 4k 81 − k 4k

For the coefficient of Tk + 1 > Tk Tk + 1 >1 the coefficient of Tk i.e. 81 − k > 1 4k 81 − k > 4k 81 > 5k 1 16 > k 5

So for k = 1, 2, 3, … 16 the coefficient of Tk + 1 > Tk For k = 17, 18, 19, … 80, the coefficient of Tk + 1 < Tk So the term with the greatest coefficient occurs when k = 16 So the greatest number of people likely to choose the TT brand is 16 16 80 − 16 (b) P (TT) = 80 C 16 c 1 m c 4 m 5 5 16 1 4 64 = 80 C 16 c m c m 5 5 = 0.11

11.9 Exercises 1. A coin is tossed 10 times. Find the probability of tossing (a) 6 tails (b) 3 tails (c) 8 heads (d) 1 head (e) at least 9 tails.

2. A coin is tossed 7 times. Find the probability of tossing (a) 1 tail (b) 6 heads (c) at least 5 heads.

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3. Kim throws 8 dice. Find the probability of throwing (a) 2 sixes (b) 5 sixes (c) 6 sixes (d) at least 7 sixes (e) fewer than 2 sixes. 4. A die is thrown 4 times. Find the probability of throwing (a) 2 threes (b) 3 ones (c) 1 four (d) less than 3 ones (e) at least 3 fives. 5. The probability of an arrow hitting a target is 0.7. If 8 arrows are fired, find the probability of hitting the target (a) 5 times (b) twice (c) 3 times (d) 7 times (e) at least 6 times. 6. The probability that dogs of a certain breed will have black 2 spots is . If a dog of this breed 5 has 5 pups, find the probability that (a) 3 will have black spots (b) none will have black spots (c) 4 will have black spots (d) at least 3 will have black spots (e) fewer than 3 will have black spots. 7. The probability of a certain brand of light globe being faulty is 1.3%. In a batch of 8 such light globes, find the probability, as a percentage correct to 1 decimal place, that (a) none will be faulty (b) 6 will work (c) at least 7 will work.

8. A survey showed that about 70% of people liked Crunchy Muesli for breakfast. If another survey is carried out with 20 people, find the probability that (a) 12 people eat Crunchy Muesli (b) 13 people eat Crunchy Muesli (c) 9 people eat Crunchy Muesli. 9. The probability on average of a 3 traffic light being red is . If there 7 are 8 traffic lights on my way to work, find the probability that (a) 7 will be red (b) 6 will be red (c) more than 5 will be red (d) 2 will be red (e) they will be all green. 10. If the probability of a manufacturer’s machine breaking 3 down is , find the probability 11 that at any particular time 7 machines out of a total of 20 machines will be broken down (leave your answer in index form). 11. A photocopy machine has a 2 probability of of breaking down 5 at any one time. (a) A business owns 2 of these machines. Find the probability that at any one time (i) 1 machine is broken down (ii) at least 1 machine is broken down. (b) Another business owns 10 of these machines. Find the probability, correct to 2 decimal places, that at any one time (i) 4 of these machines are broken down (ii) less than 3 machines are broken down.

Chapter 11 Probability

12. (a) A treadmill in a gym has a 4 probability of of being used out 7 of peak times. If a gym has 9 treadmills, find the probability (to 2 decimal places) that 6 are being used out of peak times. (b) A stepper has a probability 4 of of being used out of peak 9 times. If a gym has 6 steppers, find the probability that at least 5 are being used out of peak times. (c) Find the probability that 3 treadmills and 2 steppers are being used out of peak times. (Answer correct to 2 decimal places.) 13. A casino has 20 poker machines and 8 roulette wheels. The probability of a poker machine 1 being faulty is and the 6 probability of a roulette wheel 3 being faulty is . Find the 11 probability that at a certain time 3 poker machines and 2 roulette wheels are faulty (leave in index form). 14. A school owns 12 data projectors and 8 smart boards. The probability of a data projector being used in a certain period is 3 and the probability of a smart 4 board being used in the same 5 period is . Find the probability 6 that 3 data projectors and 5 smart boards are being used at this time (in index form). 15. A quiz has 20 multiple choice questions, each with a choice of 4 answers. If Jay guesses an answer for each question at random, find the probability that he will score 75% on this quiz, correct to 2 significant figures.

16. A library has 10 different books on ancient history. If 70 people borrow one of these books at random over a period of months, what is the greatest number of people likely to borrow the book Catullus and Co? 17. Silvana rolls a die 50 times. (a) What is the greatest number of 6’s is she likely to roll? (b) Find the probability of Silvana rolling this number of 6s (leave in index form). 18. Phuong takes the 4 aces from a deck of cards and randomly selects an ace out of these 4 cards over 100 experiments.

(a) What is the greatest number of times that he is likely to have selected the ace of hearts? (b) Find the probability that he selects the ace of hearts this number of times (to 3 significant figures).

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19. One bag contains 5 blue and 3 yellow balls and another bag contains 7 blue and 2 yellow balls. (a) A ball is drawn out at random from each bag. Find the probability of getting (i) 2 blue balls (ii) one yellow and one blue ball (iii) at least one yellow ball. (b) A bag is chosen at random and a ball drawn out. What is the probability that it is blue? (c) A bag is chosen at random and a ball drawn out. The result is recorded, then the ball is placed back in the bag. If this is done 20 times, find the probability of drawing out 12 blue balls correct to 3 decimal places.

20. One bag contains 3 black and 4 white balls and another bag contains 5 black and 3 white balls. (a) A bag is chosen at random and a ball drawn out. What is the probability that it is white? (b) A bag is chosen at random and a ball drawn out. The result is recorded, then the ball is placed back in the bag. If this is done 15 times, find the probability of drawing out 9 white balls (to 3 decimal places).

Chapter 11 Probability

Test Yourself 11 1. The probability that a certain type of seed will germinate is 93%. If 3 of this type of seeds are planted, find the probability that (a) all will germinate (b) just 1 will germinate (c) at least 1 will germinate. 2. A game is played where the differences of the numbers on 2 dice are taken. (a) Draw a table showing the sample space (all possibilities). (b) Find the probability of getting a difference of (i) 3 (ii) 0 (iii) 1 or 2. 3. Mark buys 5 tickets in a raffle in which 200 are sold altogether. (a) What is the probability that he will (i) win (ii) not win the raffle? (b) If the raffle has 2 prizes, find the probability that Mark will win just 1 prize.

6. A set of 100 cards numbered 1 to 100 is placed in a box and one drawn at random. Find the probability that the card chosen will be (a) odd (b) less than 30 (c) a multiple of 5 (d) less than 30 or a multiple of 5 (e) odd or less than 30.

4. In a class of 30 students, 17 study history, 11 study geography and 5 study neither. One of these students is chosen at random. Find the probability that this student will (a) study geography but not history (b) study both history and geography.

9. There are 7 different colours and 8 different sizes of leather jackets in a shop. If Jean selects a jacket at random, find the probability that she will select one the same size and colour as her friend does.

5. ‘In the casino, when tossing 2 coins, 2 tails came up 10 times in a row. So there is less chance that 2 tails will come up next time.’ Is this statement true? Why?

3 of 5 winning and a second game has a 2 probability of of winning. If Jenny 3 plays one of each game, find the probability that she wins (a) both games (b) one game (c) neither game.

7. One game has a probability of

8. A bag contains 5 black and 7 white marbles. Two are chosen at random from the bag (a) with replacement (b) without replacement. Find the probability of getting a black and a white marble.

10. Each of a certain type of machine in a factory has a probability of 4.5% of breaking down at any time. If the factory has 3 of these machines, find the probability that at any one time (a) all will be broken down (b) at least one will be broken down.

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11. In how many ways can a 3 digit number be made at random from the digits 1, 2, 3, 4 and 5 if (a) no digit may be used twice? (b) the digits may be repeated and the number is greater than 300? 12. A bag contains 4 yellow, 3 red and 6 blue balls. Two are chosen at random. Find the probability of choosing (a) 2 yellow balls (b) a red and a blue ball (c) 2 blue balls. 13. The probability that advertisements are showing when changing TV channels 2 at random is . If George changes 5 channels 6 times during an evening, find the probability that advertisements are showing (a) on 2 occasions (b) more than 4 times. 14. In a class of 25 students, 4 are selected at random to attend a workshop. (a) What is the probability that 4 friends are chosen? (b) If there are 16 females and 9 males in the class, in how many ways can 2 females and 2 males be selected for the workshop? 15. In a group of 12 friends, 8 have seen the movie Star Wars 20 and 9 have seen the movie Mission Impossible 6. Everyone in the group has seen at least one of these movies. If one of the friends is chosen at random, find the probability that this person has seen (a) both movies (b) only Mission Impossible 6.

16. Twelve people are to be seated around a table. (a) In how many ways can they be seated? (b) In how many ways can they be seated if 2 particular people are not to be put together? (c) Find the probability that 2 friends will be seated together. 17. A game of chance has a

2 probability of 5

3 probability of a draw. 8 (a) If I play one of these games, find the probability of losing. (b) If I play 2 of these games, find the probability of (i) a win and a draw (ii) a loss and a draw (iii) 2 wins.

a win or a

18. A card is chosen at random from a set of 10 cards numbered 1 to 10. A second card is chosen from a set of 20 cards numbered 1 to 20. Find the probability that the combination number these cards make is (a) 911 (b) less than 100 (c) between 300 and 500. 2 probability of 3 coming up 6. The other numbers have an equal probability of coming up. If the die is rolled, find the probability that it comes up (a) 2 (b) even.

19. A loaded die has a

20. In how many ways can the word AUSTRALIA be arranged?

Chapter 11 Probability

21. Amie buys 3 raffle tickets. If 150 tickets are sold altogether, find the probability that Amie wins (a) 1st prize (b) only 2nd prize (c) 1st and 2nd prizes (d) neither prize. 22. A bag contains 6 white, 8 red and 5 blue balls. If two balls are selected at random, find the probability of choosing a red and a blue ball (a) with replacement (b) without replacement. 23. A group of 9 friends go to the movies. If 5 buy popcorn and 7 buy ice creams, find the probability that one friend chosen at random will have

(a) popcorn but not ice cream (b) both popcorn and ice cream. 24. The probability that an arrow will hit a 8 target is . 9 (a) If 3 arrows are fired, find the probability that (i) 2 hit the target (ii) at least 1 hits the target. (b) If 12 arrows are fired, find the probability that 5 hit the target (to 2 significant figures). 3 5 and the probability of winning Game B is 7 . Find the probability of winning 10 (a) both games (b) neither game (c) one game.

25. The probability of winning Game A is

Challenge Exercise 11 1. In a group of 35 students, 25 go to the movies and 15 go to the football. If all the students like at least one of these activities, find the probability that a student chosen at random will (a) go to both the movies and the football (b) only go to the movies. 2. In a train compartment, there are 8 seats, with 4 facing the front and 4 facing backwards. (a) If 5 people sit in the compartment, in how many ways can they be arranged? (b) If 2 of the people do not like sitting

backwards, in how many ways can the 5 people be arranged? (c) Find the probability that 2 particular people will sit opposite each other if seating is arranged at random. 3. In how many different ways can the word MISSISSIPPI be arranged? 4. A certain soccer team has a probability of 0.5 of winning a match and a probability of 0.2 of drawing. If the team plays 2 matches, find the probability that it will (a) draw both matches (b) win at least 1 match (c) not win either match.

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5. A game of poker uses a deck of 52 cards with 4 suits (hearts, diamonds, spades and clubs). Each suit has 13 cards, consisting of an ace, cards numbered from 2 to 10, a jack, queen and king. If a person is dealt 5 cards find the probability of getting (a) four aces (b) a flush (all cards the same suit). 6. I throw a coin k times. Find an expression to describe the probability of throwing (a) at least 1 tail (b) (k − 3) heads (c) 9 tails. 7. The probability of an egg hatching out at a certain farm is 73%. (a) If there are 20 eggs, find the probability that they will all hatch. (b) Find the probability that 17 eggs will hatch. (c) Find the probability that k eggs will hatch. 8. Twelve students sit at a round table. (a) How many ways can they be arranged? (b) If 4 students wish to sit together, how many seating arrangements can be made? (c) Find the probability that 2 friends will be separated from each other if the seating arrangement is random. 9. A boat has 2 seats facing the bow and 2 seats facing aft. Four people are sitting in the boat. (a) How many ways can they be arranged in the seats? (b) One person does not like to sit facing aft. How many ways can the seating be arranged?

10. A squad of 8 is chosen at random from 3 baseball teams with 10 players in each team. (a) In how many ways can this squad be selected? (b) If 5 of the squad are chosen from the A team, and 2 from the B team, and 1 is chosen from the C team, how many ways can the squad be formed? (c) Find the probability that Joe from the B team and Dan from the A team will be chosen. 11. There are n seats around a circular table, and n people are arranged randomly around the table. (a) In how many ways can they be arranged? (b) What is the probability of 2 particular people sitting together? (c) Show that the probability of 3 particular people sitting together is 6 . (n − 1) (n − 2) (d) What is the probability of k people sitting together? 12. If a card is drawn out at random from a set of playing cards find the probability that it will be (a) an ace or a heart (b) a diamond or an odd number (c) a jack or a spade. 13. Bill does not select the numbers 1, 2, 3, 4, 5 and 6 for Lotto as he says this combination would never win. Is he correct?

Chapter 11 Probability

14. In a set of 5 cards, each has one of the letters A, B, C, D and E on it. If two cards are selected at random with replacement, find the probability that (a) both cards are A’s (b) one card is an A and the other is a D (c) neither card is an A or D.

17. A game involves tossing 2 coins and rolling 2 dice. The scoring is shown in the table.

15. In the game of Yahtzee, 5 dice are rolled. Find the probability of rolling (a) all 6’s (b) all the same number (c) 3 6’s 16. Out of a class of 30 students, 19 play a musical instrument and 7 play both a musical instrument and a sport. Two students play neither. (a) One student is selected from the class at random. Find the probability that this person plays a sport but not a musical instrument. (b) Two people are selected at random from the class. Find the probability that both these people only play a sport.



Result

Score (points)

2 heads and double 6

5

2 heads and double (not 6)

3

2 tails and double 6

4

2 tails and double (not 6)

2

(a) Find the probability of getting 2 heads and a double 6. (b) Find the probability of getting 2 tails and a double that is not 6. (c) What is the probability that Andre will score 13 in three moves? (d) What is the probability that Justin will beat Andre’s score in three moves?

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