IB Maths worksheet Probability

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

(i) exactly once?

(ii) at least once?

(b) The process of selecting and replacing is carried out 400 times.

What is the expected number of black discs that would be drawn?

2. A new blood test has been shown to be effective in the early detection of a disease. The

probability that the blood test correctly identifies someone with this disease is 0.99, and

the probability that the blood test correctly identifies someone without that disease is

0.95. The incidence of this disease in the general population is 0.0001.

A doctor administered the blood test to a patient and the test result indicated that this

patient had the disease. What is the probability that the patient has the disease?

3. The quality control department of a company making computer chips knows that 2% of

the chips are defective. Use the normal approximation to the binomial probability

distribution, with a continuity correction, to find the probability that, in a batch

containing 1000 chips, between 20 and 30 chips (inclusive) are defective.

4. A supplier of copper wire looks for flaws before despatching it to customers.

It is known that the number of flaws follow a Poisson probability distribution

with a mean of 2.3 flaws per metre.

(a) Determine the probability that there are exactly 2 flaws in 1 metre of the wire.

(b) Determine the probability that there is at least one flaw in 2 metres of the wire.

5. 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.

6. The local Football Association consists of ten teams. Team A has a 40 % chance of

winning any game against a higher-ranked team, and a 75 % chance of winning any

game against a lower-ranked team. If A is currently in fourth position, find the

probability that A wins its next game.

7. The continuous random variable X has probability density function f (x) where

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fk(x) =

otherwise,0

10,ee xk kx

(a) Show that k = 1.

(b) What is the probability that the random variable X has a value that lies between

41 and

21 ? Give your answer in terms of e.

(c) Find the mean and variance of the distribution. Give your answers exactly, in

terms of e.

The random variable X above represents the lifetime, in years, of a certain type of

battery.

(d) Find the probability that a battery lasts more than six months. A calculator is

fitted with three of these batteries. Each battery fails independently of the other

two. Find the probability that at the end of six months

(e) none of the batteries has failed;

(f) exactly one of the batteries has failed.

8. 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 Total

Television

Sport 33 29

Total 46 100

By completing this table or otherwise, find the probability that

(a) a student selected at random prefers to watch television;

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

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

U A B

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

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

(b) Find

(i) n(A B);

(ii) P(A B).

(c) Explain why events A and B are not mutually exclusive.

10. Given that events A and B are independent with P(A B) = 0.3 and P(A B) = 0.3,

find P(A B). (Total 3 marks)

11. A satellite relies on solar cells for its power and will operate provided that at least one of

the cells is working. Cells fail independently of each other, and the probability that an

individual cell fails within one year is 0.8.

(a) For a satellite with ten solar cells, find the probability that all ten cells fail within

one year.

(b) For a satellite with ten solar cells, find the probability that the satellite is still

operating at the end of one year.

(c) For a satellite with n solar cells, write down the probability that the satellite is still

operating at the end of one year. Hence, find the smallest number of solar cells

required so that the probability of the satellite still operating at the end of one year

is at least 0.95.

12. The lifetime of a particular component of a solar cell is Y years, where Y is a continuous

random variable with probability density function

0. when e5.0

0 when 0)(

/2- y

yyf

y

(a) Find the probability, correct to four significant figures, that a given component

fails within six months.

Each solar cell has three components which work independently and the cell will

continue to run if at least two of the components continue to work.

(b) Find the probability that a solar cell fails within six months.

13. A girl walks to school every day. If it is not raining, the probability that she is late is 5

1.

If it is raining, the probability that she is late is 3

2. The probability that it rains on a

particular day is 4

1.

On one particular day the girl is late. Find the probability that it was raining on that day.

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14. Given that ,4

1)(P)(P,

3

2)(P XYXYX , find

(a) P(Y);

(b) P(X Y).

15. In a game, the probability of a player scoring with a shot is 4

1. Let X be the number of

shots the player takes to score, including the scoring shot. (You can assume that each

shot is independent of the others.)

(a) Find P(X = 3).

(b) Find the probability that the player will have at least three misses before scoring

twice.

(c) Prove that the expected value of X is 4.

(You may use the result (1 – x)–2 = 1 + 2x + 3x2 + 4x3......)

16. (a) Patients arrive at random at an emergency room in a hospital at the rate of 15 per

hour throughout the day. Find the probability that 6 patients will arrive at the

emergency room between 08:00 and 08:15.

(b) The emergency room switchboard has two operators. One operator answers calls

for doctors and the other deals with enquiries about patients. The first operator

fails to answer 1% of her calls and the second operator fails to answer 3% of his

calls. On a typical day, the first and second telephone operators receive 20 and 40

calls respectively during an afternoon session. Using the Poisson distribution find

the probability that, between them, the two operators fail to answer two or more

calls during an afternoon session.

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

A

BU

(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) Find n(B A').

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(c) An element is selected at random from U. What is the probability that this element

is

in B A?

(Total 4 marks) 18. 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.

0.4 C

C

B

B

B

B

BiologyChemistry

(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?

19. A continuous random variable X has probability density function

elsewhere

,10for,

,0

)1(

4

)(2

xxxf

Find E(X).

20. Two women, Ann and Bridget, play a game in which they take it in turns to throw an

unbiased six-sided die. The first woman to throw a '6' wins the game. Ann is the first to

throw.

(a) Find the probability that

(i) Bridget wins on her first throw;

(ii) Ann wins on her second throw;

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(iii) Ann wins on her nth throw.

(b) Let p be the probability that Ann wins the game. Show that .36

25

6

1pp

(c) Find the probability that Bridget wins the game.

(d) Suppose that the game is played six times. Find the probability that Ann wins

more games than Bridget.

21. A box contains 22 red apples and 3 green apples. Three apples are selected at random,

one after the other, without replacement.

(a) The first two apples are green. What is the probability that the third apple is red?

(b) What is the probability that exactly two of the three apples are red?

22. The probability that it rains during a summer‟s day in a certain town is 0.2. In this town,

the probability that the daily maximum temperature exceeds 25 °C is 0.3 when it rains

and 0.6 when it does not rain. Given that the maximum daily temperature exceeded 25

°C on a particular summer‟s day, find the probability that it rained on that day.

23. When John throws a stone at a target, the probability that he hits the target is 0.4. He

throws a stone 6 times.

(a) Find the probability that he hits the target exactly 4 times.

(b) Find the probability that he hits the target for the first time on his third throw.

24. Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The

score for each child is the sum of the two numbers shown on their respective dice.

(a) (i) Calculate the probability that Alan obtains a score of 9.

(ii) Calculate the probability that Alan and Belle both obtain a score of 9.

(b) (i) Calculate the probability that Alan and Belle obtain the same score,

(ii) Deduce the probability that Alan‟s score exceeds Belle‟s score.

(c) Let X denote the largest number shown on the four dice.

(i) Show that for P(X x) = 4

6

x, for x = 1, 2,... 6

(ii) Copy and complete the following probability distribution table.

x 1 2 3 4 5 6

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P(X = x) 1296

1 1296

15

1296

671

(iii) Calculate E(X).

25. The probability density function f(x), of a continuous random variable X is defined by

f(x) =

otherwise.,0

20),–4(4

1 2 xxx

Calculate the median value of X.

26. (a) At a building site the probability, P(A), that all materials arrive on time is 0.85.

The probability, P(B), that the building will be completed on time is 0.60. The

probability that the materials arrive on time and that the building is completed on time is

0.55.

(i) Show that events A and B are not independent.

(ii) All the materials arrive on time. Find the probability that the building will

not be completed on time.

(b) There was a team of ten people working on the building, including three

electricians and two plumbers. The architect called a meeting with five of the

team, and randomly selected people to attend. Calculate the probability that

exactly two electricians and one plumber were called to the meeting.

(c) The number of hours a week the people in the team work is normally distributed

with a mean of 42 hours. 10% of the team work 48 hours or more a week. Find the

probability that both plumbers work more than 40 hours in a given week.

27. Give your answers to four significant figures.

A machine produces cloth with some minor faults. The number of faults per metre is a

random variable following a Poisson distribution with a mean 3. Calculate the

probability that a metre of the cloth contains five or more faults.

28. 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).

(a) P(A|B) = P(A)

(b) P(A B) = 0

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(c) P(A B) = P(A)

29. 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.

a b c

E(32) H(28)

(a) Calculate the values a, b, c.

(b) A student is selected at random.

(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.

(c) 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.

30. The independent events A, B are such that P(A) = 0.4 and P(A B) = 0.88. Find

(a) P(B)

(b) the probability that either A occurs or B occurs, but not both.

31. Give all numerical answers to this question correct to three significant figures.

Two typists were given a series of tests to complete. On average, Mr Brown made 2.7

mistakes per test while Mr Smith made 2.5 mistakes per test. Assume that the number of

mistakes made by any typist follows a Poisson distribution.

(a) Calculate the probability that, in a particular test,

(i) Mr Brown made two mistakes;

(ii) Mr Smith made three mistakes;

(iii) Mr Brown made two mistakes and Mr Smith made three mistakes.

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(b) In another test, Mr Brown and Mr Smith made a combined total of five mistakes.

Calculate the probability that Mr Brown made fewer mistakes than Mr Smith.

32. 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.

33. Dumisani is a student at IB World College.

The probability that he will be woken by his alarm clock is 8

7.

If he is woken by his alarm clock the probability he will be late for school is 4

1.

If he is not woken by his alarm clock the probability he will be late for school is 5

3.

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.

(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?

34. Robert travels to work by train every weekday from Monday to Friday. The probability

that he catches the 08.00 train on Monday is 0.66. The probability that he catches the

08.00 train on any other weekday is 0.75. A weekday is chosen at random.

(a) Find the probability that he catches the train on that day.

(b) Given that he catches the 08.00 train on that day, find the probability that the

chosen day is Monday.

35. Jack and Jill play a game, by throwing a die in turn. If the die shows a 1, 2, 3 or 4, the

player who threw the die wins the game. If the die shows a 5 or 6, the other player has

the next throw. Jack plays first and the game continues until there is a winner.

(a) Write down the probability that Jack wins on his first throw.

(b) Calculate the probability that Jill wins on her first throw.

(c) Calculate the probability that Jack wins the game.

36. Let f(x) be the probability density function for a random variable X, where

f(x)

otherwise,0

20for,2 xkx

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(a) Show that k = 8

3.

(b) Calculate

(i) E(X);

(ii) the median of X.

37. The random variable X has a Poisson distribution with mean λ. Let p be the probability

that X takes the value 1 or 2.

(a) Write down an expression for p.

(b) Sketch the graph of p for 0 λ 4.

(c) Find the exact value of λ for which p is a maximum.

38. 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.

Red

Yellow

Does not grow

Does not grow

Grows

Grows

0.4

0.9

(b) (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.

39. Let X be a random variable with a Poisson distribution such that Var(X) = (E(X))2 – 6.

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(a) Show that the mean of the distribution is 3.

(b) Find P(X 3).

Let Y be another random variable, independent of X, with a Poisson distribution such

that E(Y) = 2.

(c) Find P(X + Y < 4).

(d) Let U = X + 2Y.

(i) Find the mean and variance of U.

(ii) State with a reason whether or not U has a Poisson distribution.

40. 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.

41. The speeds of cars at a certain point on a straight road are normally distributed with

mean and standard deviation . 15 % of the cars travelled at speeds greater than 90 km

h–1 and 12 % of them at speeds less than 40 km h–1. Find and .

42. 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

X 0 1 2

P(X = x)

10

3

10

6

10

1

(a) 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.

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(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.

43. The random variable X has a Poisson distribution with mean 4. Calculate

(a) P(3 X 5);

(b) P(X 3);

(c) P(3 X < 5X 3).

44. Bag A contains 2 red and 3 green balls.

(a) Two balls are chosen at random from the bag without replacement. Find the

probability that 2 red balls are chosen.

Bag B contains 4 red and n green balls.

(b) Two balls are chosen without replacement from this bag. If the probability that

two red balls are chosen is 15

2, show that n = 6.

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.

(c) Calculate the probability that two red balls are chosen.

(d) Given that two red balls are chosen, find the probability that a 1 or a 6 was

obtained on the die.

45. The continuous random variable X has probability density function

f(x) = 6

1x(1 + x2) for 0 x 2,

f(x) = 0 otherwise.

(a) Sketch the graph of f for 0 x 2.

(b) Write down the mode of X.

(c) Find the mean of X.

(d) Find the median of X.

46. When a fair die is thrown, the probability of obtaining a „6‟ is 6

1.

Charles throws such a die repeatedly.

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(a) Calculate the probability that

(i) he throws at least two „6‟s in his first ten throws;

(ii) he throws his first „6‟ on his fifth throw;

(iii) he throws his third „6‟ on his twelfth throw.

(b) On which throw is he most likely to throw his first „6‟?

47. 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.

Find

(a) P(E);

(b) P(F);

(c) P(E F).

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

Year 1 Year 2 Totals

History 50 35 85

Science 15 30 45

Art 45 35 80

Totals 110 100 210

(a) 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. 49. A team of five students is to be chosen at random to take part in a debate. The team is to

be chosen from a group of eight medical students and three law students. Find the

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probability that

(a) only medical students are chosen;

(b) all three law students are chosen.

50. The probability density function f(x) of the continuous random variable X is defined on

the interval [0, a] by

. 3for 8

27

.3 0for 81

)(

2ax

x

xxxf

Find the value of a.

51. Given that (A B) = , P(A|B) = 31 and P(A) =

76 , find P(B).