Chapter1 Exercises

1 CHAPTER 1 EXERCISES 1.

A 'word' is made by writing the seven letters used to spell 'EXAMPLE'

in some order. Find how many different 'words' are possible in each of the following cases. a. The first letter and the last letter are each 'E'. b. The two letters 'E' are next to each other. c. The two letters 'E' are not next to each other. 2.

Find the number of ways in which 4 questions can be chosen from the 7

questions in Section 2 of this examination paper, assuming that the order in which the questions are chosen is not relevant. 3.

A delegation of 3 girls and 2 boys is to be selected from a class of 18 girls

and 12 boys. Find the number of possible delegation. 4.

A computer terminal displaying text can generate 16 different colours

numbered 1 to 16. Any one of colours 1 to 8 may be used as the "background colour" on the screen, and any one of colours 1 to 16 may be used as the "text colours"; however, selecting the same colour for background and text renders the text invisible and so this combination is not used. Find the number of different usable combinations of background colour and text colour. 5.

A delegation of four pupils is to be chosen from a class of four boys and

five girls. Find the number of such delegations that contain at least one boy and at least one girl. 6.

A code consists of blocks of ten digits, four of which are zeros and six

of which are ones; e.g. 1011011100. Calculate the number of such blocks in which the first and the last digits are the same as each other. 7.

A nursery school teacher has 4 apples, 3 oranges and 2 bananas to share

among 9 children with each child receiving one fruit. Find the number of different ways in which this can be done. 8.

a. A tennis club has n male players and n female players. For a tournament the players are to be arranged in n pairs, with each pair consisting of one male and one female. Find the number of possible pairings.

2 b. In the state of Utopia, the alphabet contains 25 letters. A car registration number consists of two different letters of the alphabet followed by an integer n such that 100 6 n 6 999. Find the number of possible car registration numbers. 9.

A set of 20 students is made up of 10 students from each of two different

year-groups. Five students are to be selected from the set, and the order of selection is unimportant. Find a. the total number of possible selection. b. the number of selection in which there are at least two students from each of the two year-groups. 10.

A panel of judges in an essay competition ha to select, and place in order

of merit, 4 winning entries from a total entry of 20. Find the number of ways in which this can be done. As a first step in the selection, 5 finalists are selected, without being placed in order. Find the number of ways in which this can be done. 11.

Find the number of arrangements of all nine letters of the word SELEC-

TION in which a. the two letters E are next to each other. b. the two letters E are not next to each other. 12.

Eleven cards each bear a single letter, and together they can be made to

spell the word "EXAMINATION". Three cards are selected from the eleven cards, and the order of selection is not relevant. Find how many possible selections can be made a.

if the three cards all bear different letters.

b.

if two of the three cards bear the same letter.

13.

Find the number of 4-letter code-words that can be made from the letters

of the word ADVANCE, a.

using neither of the "A"s,

3 b.

using both of the "A"s.

14.

In this question, a 'word' is defined to be any set of letters in a row,

whether or not it makes sense. Find how many different 'words' can be made using all 8 letters of the word SYLLABUS. 15. a.

Eight people go to the theatre and sit in a particular group of eight adjacent reserved seats in the front row. Three of the eight belong to one family and sit together. i.

If the other five people do not mind where they sit, find the number of possible seating arrangements for all eight people.

ii.

If the other five people do not mind where they sit, except that two of them refuse to sit together, find the number of possible seating arrangements for all eight people.

b.

The salad bar at a restaurant has 6 separate bowls containing lettuce, tomatoes, cucumber, radishes, spring onions and beetroot respectively. John decides to visit the salad bar and make a selection. At each bowl, he can choose to take some of the contents or not. i.

Assuming that John takes some of the contents from at least one bowl, find how many different selections he can make.

ii.

John decides he is going to have 4 salad items, and one of them will be tomatoes. How many different selections can he make?

16.

A ten-digit number is formed by writing down the digits 0, 1, 2, 3, 4,

5, 6, 7, 8, 9 in some order. No number is allowed to start with 0. Find how many such numbers are odd. 17.In

how many ways can a committee of 3 men and 3 women be chosen

from a group of 7 men and 6 women? The oldest of the 7 men is A and the oldest of the 6 women is B. It is decided that the committee can include at most one of A and B. In how many ways can the committee now be chosen?

4 18.

You have a handphone with Vinaphone sevice. You topup your mobile

by inputing 4 series of number at random. At that time there are 1 million unactivated topup cards. What is the probability of success? 19.

Consider the following situation. There are 3 people, A, B and C, in a

room and each of these 3 people gets a hat on their head which is either red or blue at random. Everybody can see the hat of the other two people, but they can not see their own hat. The people in the room are not allowed to talk to each other. We view A, B, C as a team and we ask the team to select at least one of them who has to guess the colour of his own hat. Before the team entered the room, and before they receive their hats they can agree on a certain stratergy. The question is whether or not the team can come up with a stratergy which yields only correct guesses with probability larger than 50%. Noting that if the team make one guess, this guess should be correct, if the team decides to select two or three persons to guess their colour, all guesses must be correct for the team to win. 20.

Suppose that we have a population of people. Suppose in addition that

the probability that an individual has a certain disease is 1/10,000. There is a test for this disease and this test is 99% accurate in the sense that the probability that a sick person is tested positive is 99% and that a healthy person is tested positive with probability 0.01. One particular individual is tested positive. What is the probability that he is sick? 21.

A person has $k. He decides to play a game until going bankrupt or

getting $N (enough money to buy a car). Suppose that the probability of winning in each game is p. What is the probability that he can afford a car before bankruptcy? 22.

A circular card is divided into 3 sectors scoring 1, 2, 3 and having angles

135o , 90o , 135o , respectively. On a second circular card, sectors scoring 1, 2, 3 have angles 180o , 90o , 90o respectively. Each card has a pointer pivoted at its center. After being set in motion, the pointers come to rest independently in random positions. Find the probability that a. the score on each card is 1, b. the score on at least one of the cards is 3.

5 23.

A bag contains 4 red counters and 6 green counters. Four counters are

drawn at random from the bag, without replacement. Calculate the probability that a. all the counters drawn are green, b. at least one counter of each colour is drawn, c. at least two green counters are drawn, d. at least two green counters are drawn, given that at least one of each colour is drawn. State with a reason whether or not the events 'at least two green counters are drawn' and 'at least one counter of each colour is drawn' are independent. 24.

A choir has 7 sopranos, 6 altos, 3 tenors and 4 basses. The sopranos and

altos are women and tenors and basses are men. At a particular rehearsal, three members of the choir are chosen at random to make the tea. a. Find the probability that all three tenors are chosen. b. Find the probability that exactly one bass is chosen. c. Find the conditional probability that two women are chosen, given that exactly one bass is chosen. d. Find the probability that the chosen group contains exactly one tenor or exactly one bass (or both). 25.

A game is played, by a single player, with a set of ten cards, which are

numbered 11, 12, 13, 14,..., 20 respectively. The cards are placed in a bag, and the player takes one card, chosen at random, from the bag. If the number on this card is prime the player wins the game, and if the number is even the player loses the game. If the number on the card is 15 the player takes a second card, chosen at random from nine cards remaining in the bag; the player wins the game if the number on this second card is prime and loses the game otherwise. Find the probability that, in a particular game, a. the player wins,

6 b. only one card is taken from the bag, given that the player loses For each of the following possible amendments to the rules, find the probability that the player wins a particular game. a. The player takes two cards, chosen at random, from the bag. The player wins the game if either number (or both) is prime, and loses the game if neither number is prime. b. The player takes cards chosen at random, one by one and with replacement, from the bag, continuing until either a prime number results or an even number results. The player wins if the number on the last card chosen is prime and loses otherwise. 26.

In a computer game played by a single player, the player has to find,

within a fixed time, the path through a maze shown on the computer screen. On the first occasion that a particular player plays the game, the computer shows a simple maze, and the probability that the player succeeds in finding the path in the time allowed is 3/4. On subsequent occasions, the maze shown depends on the result of the previous game. If the player succeeded on the previous occasion, the next maze is harder, and the probability that the player succeeds is one half of the probability of success on the previous occasion. If the player failed on the previous occasion, a simple maze is shown and the probability of the player succeeding is again 3/4. The player plays three games. a. Show that the probability that the player succeeds in all three games is

27/512. b. Find the probability that the player succeeds in exactly one of the games. c. Find the probability that the player does not have two consecutive successes. d. Find the conditional probability that the player has two consecutive successes given that the player has exactly two successes.

7 27.

Ten ordinary dice are thrown, and the number of "sixes" showing as a

result is counted. Find the probability that, in one throw of the ten dice, a. exactly 3 "sixes" will show, b. more than 3 "sixes" will show. 28.

A book has 60 pages. The letter 'e' is the last letter on 15 of the pages

and the letters 's', 't', and 'd' are the last letter on 12, 9 and 6, respectively, of the pages. The last letters on each of the other pages of the book are all different from each other and none is 'e', 's', 't' or 'd'. One page out of the 60 pages is chosen at random and the last letter is observed. This process is carried out two more times. Find a. the probability that the letters obtained are 't', 'e', 'e' in that order, b. the probability that the letters obtained are 't', 'e', 'e' in any order, c. the probability that the letters obtained are 't', 'e', 'e' in that order, given that the letters obtained are 't', 'e', 'e' in some order. A page is chosen at random and then a second different page is chosen at random. Find d. the probability that at least one of the two pages ends with the letter 's', e. the probability that the two pages have the same last letter as each other. 29.

a. Two fair dice are thrown, and events A, B and C are defined as follows: A: the sum of the two scores is odd, B: At least one of the two scores is greater than 4, C: the two scores are equal. Find the probability of A, B and C. Which two of these three events are i. mutually exclusive, ii. independent,

8 b. Two players A and B regularly play each other at chess. When A has the first move in a game, the probability of A winning that game is 0.4 and the probability of B winning that game is 0.2. When B has the first move in a game, the probability of B winning that game is 0.3 and the probability of A winning that game is 0.2. Any game of chess that is not won by either player ends in a draw. i. Given that A and B toss a fair coin to decide who has first move in a game, find the probability of the game ending in a draw. ii. To make their games more enjoyable, A and B agree to change the procedure for deciding who has the first move in a game. As a result of their new procedure, the probability of A having first move in any game is p. Find the value of p which gives A and B equal chances of winning each game. 30.

In a lottery there are 24 prizes allocated at random to 24 prize-winners.

Ann, Ben and Cal are three of the prize-winners. Of the prizes, 4 are cars, 8 are bicycles and 12 are watches. Show that the probability that Ann gets a car and Ben gets either a bicycle or a watch is 10/69. Giving each answer either as a fraction as a decimal correct to 3 significant figures, find a. the probability that both Ann and Ben get cars, given that Cal gets a car. b. the probability that either Ann or Cl (or both) gets a car. c. the probability that Ann gets a car and Ben gets either a car or a bicycle. d. the probability that Ann gets a car given that Ben gets either a car or a bicycle. 31.

A study of numbers of male and female children in families in a certain

population is being carried out. a. A simple model is that each child in any family is equally likely to be male or female, and that the sex of each child is independent of the sex of any previous children in the family. Using this model calculate the probability that, in a randomly chosen family of 4 children.

9 i. there will be 2 males and 2 females, ii. there will be exactly 1 female given that there is at least one female. b. An alternative model is that the first child in any family is equally likely to be male or female, but that, for any subsequent children, the probability they will be of the same sex as the previous child is 3/5. Using this model calculate the probability that, in a randomly chosen family of 4 children, i.

all four will be of the same sex,

ii. iii. 32.

no two consecutive children will be of the same sex, there will be 2 males and 2 females.

Analysis of the purchases at a snack bar over a long period shows that

55% of ccustomers buy a hot drink, and that this figure is made up of 32% who buy coffee, 19% who buy tea and 4% who buy chocolate. No one buys two drinks. Find the probability that at least three out of a random sample of ten customers buy coffee, giving your answer correct to three places of decimals. Find the probability that a randomly chosen customer buys coffee, given that he or she does not buy tea. 33.

A bag contains 3 red balls and 3 green balls. Balls are drawn from the

bag at random, one by one and without replacement. a.

Show that the probability that the first 3 balls drawn are red is 1/20.

b.

Find the probability that the first 3 balls drawn consist of 2 red balls and 1 green ball (in any order). Hence or otherwise show that the probability that the third red ball appears on the fourth draw is 3/20.

c.

Find the probability that the third red ball appears on the fifth draw.

34.

A game is played by throwing a pair of unbiased dice, one red and one

green. The score is obtained as follows: If the red die is not show a six, the score is the number showing on the green die, if the red die shows a six, the score is twice the number showing on the green die.

10 a.

A player throws the pair of dice once. i. ii.

Show that the probability of the score being 4 or less is 11/18. Find the conditional probability that the red die does not show a six, given that the score is more than 4.

b.

A player throws the pair of dice 8 times. Find, correct to 3 significant figures, the probability that i. ii.

35.

all of the 8 scores are 4 or less. fewer than 6 of the 8 scores are 4 or less.

There are 36 people at a gathering of two families. There are 25 people

with the name Lee and 11 people with the name Chan. Of the 25 people named Lee, 4 are single men, 5 are single women and there are 8 married couples. Of the 11 people named Chan, 2 are single men, 3 are single women and there are 3 married couples. Two people are chosen at random from the gathering. a.

Show that the probability that they both have the name Lee is 10/21.

b.

Find the probability that they are married to each other.

c.

Find the probability that they both have the name Lee, given that they are married to each other.

d.

Find the probability that they are a man and a woman with the same name.

e.

Find the probability that they are married to each other, given that they are a man and a woman with the same name.

36.

A Personal Identification Number (PIN) constists of 4 digits in order, each

of which is one of the digits 0, 1, 2,..., 9. Susie has difficulty remembering her PIN. She tries to remember her PIN and writes down what she thinks it is. The probability that the first digit is correct is 0.8 and the probability that the second digit is correct is 0.86. The probability that the first two digits are both correct is 0.72. Find

11 a.

the probability that the second digit is correct given that the first digit is correct.

b.

the probability that the first digit is correct and the second digit is incorrect.

c.

the probability that the first digit is incorrect and the second digit is correct.

d.

the probability that the second digit is incorrect given that the first digit is incorrect.

The probability that all four digits are correct is 0.7. On 12 separate occasions Susie writes down independently what she thinks is her PIN. Find the probability that the number of occasion on which all four digits are correct is less than 10. 37.

A set of 30 cards is made up of cards chosen from a number of packs of

ordinary playing cards. The numbers of cards of each type are given in the following table. Spades Hearts Diamonds Clubs King

2

3

1

3

Queen 3

3

5

2

Jack 1 2 3 2 Thus, for example, there are 2 Kings of Spades and 3 Jacks of Diamonds. a.

One card is taken at random from the set. Events H, K, J are defined as follows: H: The card taken is a Heart K: The card taken is a King J: The card taken is a Jack i.

Describe in words what the event K ∪ H represents and state the probability of this event.

ii.

Describe in words what the event J 0 ∩ H 0 represents and state the probability of this event.

12 iii.

Find the conditional probability that the card is a Diamond, given that it is a King.

b.

Two cards are taken from the set, at random and without replacement. Find the probability that both cards are Jacks. Give your answer correct to 3 places of decimals.

c.

Three cards are taken from the set, at random and without replacement. Find the probability that they are three Kings or three Queens or three Jacks. Give your answer correct to 3 places of decimals.

38.

A large number of tickets are sold in a lottery. Each ticket can win either

a small prize or a large prize, but no ticket can win two prizes. 10% of tickets win a small prize and 0.1% of tickets win a large prize. a.

If Charlie has 20 tickets, find the probability that i. ii.

b.

he wins at least 1 prize. he wins at most 3 prizes.

If Mary has 400 tickets, use suitable approximations to find the probability that

39.

i.

she wins at most 35 small prizes.

ii.

she wins at most 2 large prizes.

In a probability experiment, three containers have the following contents. A jar contains 2 white dice and 3 black dice. A white box contains 5 red balls and 3 green balls. A black box contains 4 red balls and 3 green balls. One die is taken at random from the jar. If the die is white, two balls

are taken from the white box, at random and without replacement. If the die is black, two balls are taken from the black box, at random and without replacement. Events W and M are defined as follows. W: A white die is taken from the jar. M: One red ball and one green ball are obtained.

13 Show that P (M |W ) = 15/28. Find, giving each of your answers as an exact fraction in its lowest terms. i.

P (M ∩ W ),

ii.

P (W |M ),

iii.

P (W ∪ M )

All the dice and balls are now placed in a single container, and four objects are taken at random, each object being replaced before the next one is taken. Find the probability that one object of each colour is obtained.