ibmaths.net worksheet ibhlp1201 - baixardoc

IBMATHS.NET Worksheet IBHLP1201

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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



2

fk(x) =

otherwise,0

10,ee xkkx

(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 2xxx

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,2xkx



10

(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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