Math IB Revision Probability

1.

A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted. The disc is then replaced in the box. (a)

b)

In eight such selections, what is the probability that a black disc is selected (i)

exactly once?

(ii)

at least once?

The process of selecting and replacing is carried out 400 times. What is the expected number of black discs that would be drawn?

2.

For the events A and B, p(A) = 0.6, p(B) = 0.8 and p(A ∪ B) = 1.

Find: p(A∩B), p( A ∪ B)

3.

The graph shows a normal curve for the random variable X, with mean μ and standard deviation σ . y

A 0 It is known that p(X ≥ 12) = 0.1. (a)

1

2

x

The shaded region A is the region under the curve where x ≥ 12. Write down the area of the shaded region A.

It is also known that p(X ≤ 8) = 0.1.

4.

5.

(b)

Find the value of μ, explaining your method in full.

c)

Show that σ = 1.56 to an accuracy of three significant figures.

(d)

Find p(X ≤ 11).

A fair coin is tossed eight times. Calculate (a)

the probability of obtaining exactly 4 heads;

b)

the probability of obtaining exactly 3 heads;

(c)

the probability of obtaining 3, 4 or 5 heads.

In a survey, 100 students were asked ‘do you prefer to watch television or play sport?’ Of the 46 boys in the survey, 33 said they would choose sport, while 29 girls made this choice. Boys

Girls

Sport

33

29

Total

46

Total

Television

100 1

By completing this table or otherwise, find the probability that

2

6.

(a)

a student selected at random prefers to watch television;

(b)

a student prefers to watch television, given that the student is a boy.

The lifespan of a particular species of insect is normally distributed with a mean of 57 hours and a standard deviation of 4.4 hours. (a)

The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve.

a 0

(b)

7.

b

(i)

Write down the values of a and b.

(ii)

Find the probability that the lifespan of an insect of this species is (a)

more than 55 hours;

(b)

between 55 and 60 hours.

90% of the insects die after t hours. (i)

Represent this information on a standard normal curve diagram, similar to the one given in part (a), indicating clearly the area representing 90%.

(ii)

Find the value of t.

Two ordinary, 6-sided dice are rolled and the total score is noted. (a)

Complete the tree diagram by entering probabilities and listing outcomes. O u t c o m e s . . . . . . . . . . . . . . .

6 . . . . . . . 6 . . . . . . . . . . . . . .

n

o

. . . . . . . . . . . . . . .

6 . . . . . . .

. . . . . . . n

o

t

6 . . . . . . .

(b)

t . .6 . . . . . . . . . . . . .

n

o

t . .6 . . . . . . . . . . . . .

Find the probability of getting one or more sixes. 3

8.

The following Venn diagram shows a sample space U and events A and B.

U

A

B

n(U) = 36, n(A) = 11, n(B) = 6 and n(A ∪ B)’ = 21.

9.

(a)

On the diagram, shade the region (A ∪ B)’.

(b)

Find: (i)

(c)

Explain why events A and B are not mutually exclusive.

n(A ∩ B);

(ii)

P(A ∩ B).

–1

An urban highway has a speed limit of 50 km h . It is known that the speeds of vehicles travelling on the –1 highway are normally distributed, with a standard deviation of l0 km h , and that 30% of the vehicles using the highway exceed the speed limit. (a)

–1

Show that the mean speed of the vehicles is approximately 44.8 km h .

The police conduct a ‘Safer Driving’ campaign intended to encourage slower driving, and want to know whether the campaign has been effective. It is found that a sample of 25 vehicles has a mean speed of 41.3 –1 km h . (b) Given that the null hypothesis is H0: the mean speed has been unaffected by the campaign state H1, the alternative hypothesis.

10.

(c)

State whether a one-tailed or two-tailed test is appropriate for these hypotheses, and explain why.

(d)

Has the campaign had significant effect at the 5% level?

In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20 males. (a)

Using this information, complete the table below. Males

Females

Totals

Unemployed Employed Totals (b)

11.

200

If a person is selected at random from this group of 200, find the probability that this person is (i)

an unemployed female;

(ii)

a male, given that the person is employed.

A bag contains 10 red balls, 10 green balls and 6 white balls. Two balls are drawn at random from the bag 4

without replacement. What is the probability that they are of different colours? 12.

The table below represents the weights, W, in grams, of 80 packets of roasted peanuts. Weight (W)

80 < W ≤ 85

85 < W ≤ 90

90 < W ≤ 95

95 < W ≤ 100

100 < W ≤ 105

105 < W ≤ 110

Number of packets

5

10

15

26

13

7

110 < W ≤ 115

4

(a)

Use the midpoint of each interval to find an estimate for the standard deviation of the weights.

(b)

Copy and complete the following cumulative frequency table for the above data.

Weight (W)

W≤ 85

W≤ 90

Number of packets

5

15

(c)

W≤ 95

W≤ 100

W ≤ 105

W ≤ 110

W ≤ 115 80

A cumulative frequency graph of the distribution is shown below, with a scale 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis.

5

8

0

7

0

6

0

5

0 m

N

u o f p a c k 4 0

3

0

2

0

1

0

8

b

e r

e t s

0

8

5

9

0

9

5 W

1 e i g

0 h

0 t

( g

1 0 5 r a m

1 1

0

1 1

5

s )

Use the graph to estimate (i)

the median;

(ii)

the upper quartile (that is, the third quartile).

Give your answers to the nearest gram. (d)

Let W , W , ..., W be the individual weights of the packets, and let W be their mean. What is the 1

2

80

value of the sum (W1 – W ) + (W2 – W ) + (W3 – W ) + . . . + (W79 – W ) + (W80 – W ) ? (e) 13.

One of the 80 packets is selected at random. Given that its weight satisfies 85 < W ≤ 110 , find the probability that its weight is greater than 100 grams.

Intelligence Quotient (IQ) in a certain population is normally distributed with a mean of 100 and a standard deviation of 15. 6

14.

(a)

What percentage of the population has an IQ between 90 and 125?

(b)

If two persons are chosen at random from the population, what is the probability that both have an IQ greater than 125?

(c)

The mean IQ of a random group of 25 persons suffering from a certain brain disorder was found to be 95.2. Is this sufficient evidence, at the 0.05 level of significance, that people suffering from the disorder have, on average, a lower IQ than the entire population? State your null hypothesis and your alternative hypothesis, and explain your reasoning.

The following Venn diagram shows the universal set U and the sets A and B. U

B A

(a)

Shade the area in the diagram which represents the set B ∩ A'.

n(U) = 100, n(A) = 30, n(B) = 50, n(A ∪ B) = 65. (b) (c)

15.

Find n(B ∩ A'). An element is selected at random from U. What is the probability that this element is in B ∩ A′ ?

The events B and C are dependent, where C is the event "a student takes Chemistry", and B is the event "a student takes Biology". It is known that P(C) = 0.4, P(B | C) = 0.6, P(B | C′ ) = 0.5. (a)

Complete the following tree diagram. C

h

e m

i s t r y B

i o

l o

g

y

B 0

. 4

C B′ B C′ B′

7

16.

(b)

Calculate the probability that a student takes Biology.

(c)

Given that a student takes Biology, what is the probability that the student takes Chemistry?

Bags of cement are labelled 25 kg. The bags are filled by machine and the actual weights are normally distributed with mean 25.7 kg and standard deviation 0.50 kg. (a)

What is the probability a bag selected at random will weigh less than 25.0 kg?

In order to reduce the number of underweight bags (bags weighing less than 25 kg) to 2.5% of the total, the mean is increased without changing the standard deviation. (b)

Show that the increased mean is 26.0 kg.

It is decided to purchase a more accurate machine for filling the bags. The requirements for this machine are that only 2.5% of bags be under 25 kg and that only 2.5% of bags be over 26 kg. (c)

Calculate the mean and standard deviation that satisfy these requirements.

The cost of the new machine is $5000. Cement sells for $0.80 per kg. (d) 17.

18.

Compared to the cost of operating with a 26 kg mean, how many bags must be filled in order to recover the cost of the new equipment? Two fair dice are thrown and the number showing on each is noted. The sum of these two numbers is S. Find the probability that (a)

S is less than 8;

(b)

at least one die shows a 3;

(c)

at least one die shows a 3, given that S is less than 8.

The mass of packets of a breakfast cereal is normally distributed with a mean of 750 g and standard deviation of 25 g. (a)

19.

Find the probability that a packet chosen at random has mass (i)

less than 740 g;

(ii)

at least 780 g;

(iii)

between 740 g and 780 g.

(b)

Two packets are chosen at random. What is the probability that both packets have a mass which is less than 740 g?

(c)

The mass of 70% of the packets is more than x grams. Find the value of x.

3 4 For events A and B, the probabilities are P(A) = 11 , P(B) = 11 .

(a)

6 P (A ∪ B) = 11 ;

(b)

events A and B are independent.

Calculate the value of P (A ∩ B) if

8

20.

In a country called Tallopia, the height of adults is normally distributed with a mean of 187.5 cm and a standard deviation of 9.5 cm. (a)

What percentage of adults in Tallopia have a height greater than 197 cm?

(b) 21.

A standard doorway in Tallopia is designed so that 99 % of adults have a space of at least 17 cm over their heads when going through a doorway. Find the height of a standard doorway in Tallopia. Give your answer to the nearest cm. Consider events A, B such that P(A) ≠ 0, P(A) ≠ 1, P(B) ≠ 0, and P(B) ≠ 1. In each of the situations (a), (b), (c) below state whether A and B are mutually exclusive (M); independent (I); neither (N).

22.

(a) P(A|B) = P(A) (b) A family of functions is given by

P(A ∩ B) = 0

P(A ∩ B) = P(A)

(c)

2

f(x) = x + 3x + k, where k ∈ {1, 2, 3, 4, 5, 6, 7}. One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis. 23. In a school of 88 boys, 32 study economics (E), 28 study history (H) and 39 do not study either subject. This information is represented in the following Venn diagram.

E( 3

2

)

a

(a)

Calculate the values a, b, c.

(b)

A student is selected at random.

(c)

24.

H ( 2

b

8

)

c

(i)

Calculate the probability that he studies both economics and history.

(ii)

Given that he studies economics, calculate the probability that he does not study history.

A group of three students is selected at random from the school. (i)

Calculate the probability that none of these students studies economics.

(ii)

Calculate the probability that at least one of these students studies economics.

A company manufactures television sets. They claim that the lifetime of a set is normally distributed with a mean of 80 months and standard deviation of 8 months. (a)

What proportion of television sets break down in less than 72 months? 9

(b)

c)

(i)

Calculate the proportion of sets which have a lifetime between 72 months and 90 months.

(ii)

Illustrate this proportion by appropriate shading in a sketch of a normal distribution curve.

If a set breaks down in less than x months, the company replace it free of charge. They replace 4% of the sets. Find the value of x.

25.

A painter has 12 tins of paint. Seven tins are red and five tins are yellow. Two tins are chosen at random. Calculate the probability that both tins are the same colour.

26.

It is claimed that the masses of a population of lions are normally distributed with a mean mass of 310 kg and a standard deviation of 30 kg.

27.

(a)

Calculate the probability that a lion selected at random will have a mass of 350 kg or more.

(b)

The probability that the mass of a lion lies between a and b is 0.95, where a and b are symmetric about the mean. Find the value of a and of b.

Dumisani is a student at IB World College. 7 The probability that he will be woken by his alarm clock is 8 . 1 If he is woken by his alarm clock the probability he will be late for school is 4 . 3 If he is not woken by his alarm clock the probability he will be late for school is 5 . Let W be the event “Dumisani is woken by his alarm clock”. Let L be the event “Dumisani is late for school”. (a)

Copy and complete the tree diagram below. L

W

L′ L

W ′

L′ b)

Calculate the probability that Dumisani will be late for school.

(c)

Given that Dumisani is late for school what is the probability that he was woken by his alarm clock? 10

28.

A group of 75 people was asked the question “Are you in favour of banning the use of mobile phones while driving?” Their answers are shown in the following table.

Aχ

2

Yes

No

Don’t know

Men

18

10

12

Women

8

11

16

test will be used to examine the claim that answers to the question are independent of gender.

The following table gives the expected frequencies for the above data.

(a)

(b) 29.

Yes

No

Don’t know

Men

13.87

a

b

Women

12.13

c

d

(i)

Show that a = 11.2.

(ii)

Calculate the value of d.

(iii)

Find χ

2

for this data.

Explain why it is correct to conclude, at the 10 % level of significance, that responses regarding the use of mobile phones while driving do not depend on gender.

The following diagram shows a circle divided into three sectors A, B and C. The angles at the centre of the circle are 90°, 120° and 150°. Sectors A and B are shaded as shown.

A C

1

5

09 °0 1

°

2

0

°

B

11

The arrow is spun. It cannot land on the lines between the sectors. Let A, B, C and S be the events defined by A B C S 30.

: : : :

Arrow lands in sector A Arrow lands in sector B Arrow lands in sector C Arrow lands in a shaded region.

Find (a) P(B); (b) P(S); (c) P(A S). A packet of seeds contains 40 % red seeds and 60 % yellow seeds. The probability that a red seed grows is 0.9, and that a yellow seed grows is 0.8. A seed is chosen at random from the packet. (a)

Complete the probability tree diagram below. 0

0

. 4

R

31.

G

r o

w

s

D G

o e s r o w

n s

o

t

g

r o

w

D

o

n

o

t

g

r o

w

e d

Y

(b)

. 9

e l l o

w

(i)

Calculate the probability that the chosen seed is red and grows.

(ii)

Calculate the probability that the chosen seed grows.

(iii)

Given that the seed grows, calculate the probability that it is red.

e s

Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a standard deviation of 0.06 seconds. (a)

The graph below is that of the standard normal curve. The shaded area represents the probability that the reaction time of a person chosen at random is between 0.70 and 0.79 seconds.

a

0 b

12

(i)

Write down the value of a and of b.

(ii)

Calculate the probability that the reaction time of a person chosen at random is (a)

greater than 0.70 seconds;

(b)

between 0.70 and 0.79 seconds.

Three percent (3 %) of the population have a reaction time less than c seconds. (b)

32.

(i)

Represent this information on a diagram similar to the one above. Indicate clearly the area representing 3 %.

(ii)

Find c.

A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested. (a)

Write down the expected number of faulty calculators in the sample.

(b)

Find the probability that three calculators are faulty.

(c)

Find the probability that more than one calculator is faulty.

33.

The speeds of cars at a certain point on a straight road are normally distributed with mean µ and standard –1 deviation σ . 15 % of the cars travelled at speeds greater than 90 km h and 12 % of them at speeds less –1 than 40 km h . Find µ and σ .

34.

Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without replacement. Let X denote the number of red balls chosen. The following table shows the probability distribution for X

(a)

X

0

1

2

P(X = x)

3 10

6 10

1 10

Calculate E(X), the mean number of red balls chosen.

Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B. (b)

(i)

Draw a tree diagram to represent the above information, including the probability of each event.

(ii) Hence find the probability distribution for Y, where Y is the number of red balls chosen. A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B.

35.

(c)

Calculate the probability that two red balls are chosen.

(d)

Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled on the die.

Two unbiased 6-sided dice are rolled, a red one and a black one. Let E and F be the events E : the same number appears on both dice;

F : the sum of the numbers is 10.

13

Find: 36.

P(E);

(b)

P(F);

P(E ∪ F).

(c)

The table below shows the subjects studied by 210 students at a college.

(a)

37.

(a)

Year 1

Year 2

Totals

History

50

35

85

Science

15

30

Art

45

35

80

Totals

110

100

210

45

A student from the college is selected at random. Let A be the event the student studies Art. Let B be the event the student is in Year 2. (i)

Find P(A).

(ii)

Find the probability that the student is a Year 2 Art student.

(iii)

Are the events A and B independent? Justify your answer.

(b)

Given that a History student is selected at random, calculate the probability that the student is in Year 1.

(c)

Two students are selected at random from the college. Calculate the probability that one student is in Year 1, and the other in Year 2.

Residents of a small town have savings which are normally distributed with a mean of $ 3 000 and a standard deviation of $ 500. (i)

What percentage of townspeople have savings greater than $ 3 200?

(ii)

Two townspeople are chosen at random. What is the probability that both of them have savings between $ 2 300 and $ 3 300?

(iii)

The percentage of townspeople with savings less than d dollars is 74.22 %. Find the value of d.

14