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Probability

1.       A bag contains 2 red balls, 3 blue balls and 4 green balls. A ball is chosen at random from the bag and is not replaced. A second ball is chosen. Find the probability of choosing one green ball and one blue ball in any order.

Working:

 

 

Answer:

…………………………………………..

(Total 4 marks)

 

 

2.       In a bilingual school there is a class of 21 pupils. In this class, 15 of the pupils speak Spanish as their first language and 12 of these 15 pupils are Argentine. The other 6 pupils in the class speak English as their first language and 3 of these 6 pupils are Argentine.

          A pupil is selected at random from the class and is found to be Argentine. Find the probability that the pupil speaks Spanish as his/her first language.

Working:

 

 

Answer:

…………………………………………..

(Total 4 marks)

 


 

3.       A biased die with four faces is used in a game. A player pays 10 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters the player receives in return for each score.

Score

1

2

3

4

Probability

Number of counters player receives

4

5

15

n

          Find the value of n in order for the player to get an expected return of 9 counters per roll.

Working:

 

 

Answer:

…………………………………………..

(Total 4 marks)

 


 

4.       A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average weight of sugar in the bags is 1.02 kg. Assuming that the weights of the bags are normally distributed, find the standard deviation if 1.7% of the bags weigh below 1 kg.
Give your answer correct to the nearest 0.1 gram.

Working:

 

 

Answer:

…………………………………………..

(Total 4 marks)

 

 

5.       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?

(Total 6 marks)

 

 

6.       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.

(Total 7 marks)

 

 

 

 


 

7.       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.

(3)

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

(3)

(Total 6 marks)

 

 

8.       The random variable X is distributed normally with mean 30 and standard deviation 2.
Find p(27 £ X £ 34).

Working:

 

 

Answer:

....……………………………………..........

(Total 4 marks)

 


 

9.       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.

Working:

 

 

Answer:

....……………………………………..........

(Total 4 marks)

 


 

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

fk (x) =

(a)     Show that k = 1.

(3)

(b)     What is the probability that the random variable X has a value that lies between
 and ? Give your answer in terms of e.

(2)

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

(6)

          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.

(2)

          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;

(2)

(f)      exactly one of the batteries has failed.

(2)

(Total 17 marks)

 


 

11.     In a game a player rolls a biased tetrahedral (four-faced) die. The probability of each possible score is shown below.

Score

1

2

3

4

Probability

x

          Find the probability of a total score of six after two rolls.

Working:

 

 

Answer:

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

(Total 3 marks)

 


 

12.     The probability distribution of a discrete random variable X is given by

P(X = x) = k , for x = 0, 1, 2, ......

          Find the value of k.

Working:

 

 

Answer:

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

(Total 3 marks)

 

 

13.     A machine is set to produce bags of salt, whose weights are distributed normally, with a mean of 110 g and standard deviation of 1.142 g. If the weight of a bag of salt is less than 108 g, the bag is rejected. With these settings, 4% of the bags are rejected.

          The settings of the machine are altered and it is found that 7% of the bags are rejected.

(a)     (i)      If the mean has not changed, find the new standard deviation, correct to three decimal places.

(4)

          The machine is adjusted to operate with this new value of the standard deviation.

(ii)     Find the value, correct to two decimal places, at which the mean should be set so that only 4% of the bags are rejected.

(4)

(b)     With the new settings from part (a), it is found that 80% of the bags of salt have a weight which lies between A g and B g, where A and B are symmetric about the mean. Find the values of A and B, giving your answers correct to two decimal places.

(4)

(Total 12 marks)

 


 

14.     The box-and-whisker plots shown represent the heights of female students and the heights of male students at a certain school.

(a)     What percentage of female students are shorter than any male students?

(b)     What percentage of male students are shorter than some female students?

(c)     From the diagram, estimate the mean height of the male students.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(c)       ..................................................................

(Total 3 marks)

 


 

15.     Given that events A and B are independent with P(A Ç B) = 0.3 and P(A Ç B¢) = 0.3,
find P(A È B).

Working:

 

 

Answer:

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

(Total 3 marks)

 

 

16.     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.

(2)

(b)     For a satellite with ten solar cells, find the probability that the satellite is still operating at the end of one year.

(2)

(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.

(5)

(Total 9 marks)

 


 

17.     The lifetime of a particular component of a solar cell is Y years, where Y is a continuous random variable with probability density function

(a)     Find the probability, correct to four significant figures, that a given component fails within six months.

(3)

          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.

(4)

(Total 7 marks)

 

 

18.     A girl walks to school every day. If it is not raining, the probability that she is late is . If it  is raining, the probability that she is late is . The probability that it rains on a particular day is .

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

Working:

 

 

Answer:

…………………………………………..

(Total 3 marks)

 


 

19.     In a school,  of the students travel to school by bus.  Five students are chosen at random. Find the probability that exactly 3 of them travel to school by bus.

Working:

 

 

Answer:

…………………………………………..

(Total 3 marks)

 

 

20.     The diameters of discs produced by a machine are normally distributed with a mean of 10 cm and standard deviation of 0.1 cm. Find the probability of the machine producing a disc with a diameter smaller than 9.8 cm.

Working:

 

 

Answer:

…………………………………………..

(Total 3 marks)

 


 

21.     Given that P(X) =  P(Y½X) =  and P(Y½X ¢) =  find

(a)     P(Y ¢);

(b)     P(X ¢ È Y ¢).

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 

 

22.     Z is the standardized normal random variable with mean 0 and variance 1. Find the value of a such that P( | Z | £ a) = 0.75.

Working:

 

 

Answer:

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

(Total 3 marks)

 


 

23.     X is a binomial random variable, where the number of trials is 5 and the probability of success of each trial is p. Find the values of p if P(X = 4) = 0.12.

Working:

 

 

Answer:

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

(Total 3 marks)

 

 

24.     In a game, the probability of a player scoring with a shot is . 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).

(2)

 

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

(6)

 

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

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

(5)

(Total 13 marks)

 


 

25.     (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.

(3)

(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.

(5)

(Total 8 marks)

 

 

26.     A coin is biased so that when it is tossed the probability of obtaining heads is . The coin is tossed 1800 times. Let X be the number of heads obtained. Find

(a)     the mean of X;

(b)     the standard deviation of X.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 3 marks)

 


 

27.     A continuous random variable X has probability density function

          Find E(X).

Working:

 

 

Answer:

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

(Total 3 marks)

 

 

28.     The probability that a man leaves his umbrella in any shop he visits is . After visiting two shops in succession, he finds he has left his umbrella in one of them. What is the probability that he left his umbrella in the second shop?

Working:

 

 

Answer:

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

(Total 3 marks)

 

 

29.     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;

(iii)    Ann wins on her nth throw.

(6)

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

(4)

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

(2)

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

(5)

(Total 17 marks)

 

 

30.     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.

Working:

 

 

Answer:

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

(Total 6 marks)

 


 

31.     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.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

32.     The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.12 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg.

Working:

 

 

Answer:

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

(Total 6 marks)

 


 

33.     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.

(2)

(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.

(4)

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

(i)      Show that for P(X £ x) = , for x = 1, 2,... 6

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

x

1

2

3

4

5

6

P(X = x)

 

 

 

(iii)    Calculate E(X).

(7)

(Total 13 marks)

 

 

34.     The random variable X is Poisson distributed with mean m and satisfies P(X = 3) = P(X = 0) + P(X = 1).

(a)     Find the value of m, correct to four decimal places.

(3)

(b)     For this value of m evaluate P(2 £ X £ 4).

(3)

(Total 6 marks)

 


 

35.     An integer is chosen at random from the first one thousand positive integers. Find the probability that the integer chosen is

(a)     a multiple of 4;

(b)     a multiple of both 4 and 6.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

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

         f (x) =

          Calculate the median value of X.

Working:

 

 

Answer:

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

(Total 6 marks)

 

 

37.     (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.

(5)

(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.

(2)

(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.

(8)

(Total 15 marks)

 

 

38.     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.

(Total 4 marks)

 


 

39.     When a boy plays a game at a fair, the probability that he wins a prize is 0.25. He plays the game 10 times. Let X denote the total number of prizes that he wins. Assuming that the games are independent, find

(a)     E(X)

(b)     P(X £ 2).

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

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

(a)     P(B);

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

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

41.     The random variable X is normally distributed and

         P(X £ 10) = 0.670
P(X £ 12) = 0.937.

          Find E(X).

Working:

 

 

Answer:

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

(Total 6 marks)

 

 

42.     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.

(6)

 

(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.

(5)

(Total 11 marks)

 


 

43.     On a television channel the news is shown at the same time each day. The probability that Alice watches the news on a given day is 0.4. Calculate the probability that on five consecutive days, she watches the news on at most three days.

Working:

 

 

Answer:

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

(Total 6 marks)

 

 

44.     A random variable X is normally distributed with mean m and standard deviation σ, such that
P(X > 50.32) = 0.119, and P(X < 43.56) = 0.305.

(a)     Find m and s.

(5)

(b)     Hence find P(|Xm| < 5).

(2)

(Total 7 marks)

 

 

45.     The random variable X has a Poisson distribution with mean λ.

(a)     Given that P(X = 4) = P(X = 2) + P(X = 3), find the value of λ.

(3)

(b)     Given that λ = 3.2, find the value of

(i)      P(X ³ 2);

(ii)     P(X £ 3 ½ X ³ 2).

(5)

(Total 8 marks)

 

 

46.     The following diagram shows the probability density function for the random variable X, which is normally distributed with mean 250 and standard deviation 50.

 

          Find the probability represented by the shaded region.

Working:

 

 

Answer:

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

(Total 6 marks)

 


 

47.     The discrete random variable X has the following probability distribution.

P(X = x) =

          Calculate

(a)     the value of the constant k;

(b)     E(X).

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

48.     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.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

49.     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.

(1)

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

(2)

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

(3)

(Total 6 marks)

 


 

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

f (x)

(a)     Show that k = .

(2)

(b)     Calculate

(i)      E(X);

(ii)     the median of X.

(6)

(Total 8 marks)

 

 

51.     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.

(1)

(b)     Sketch the graph of p for 0 £ λ £ 4.

(1)

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

(5)

(Total 7 marks)

 


 

52.     A continuous random variable X has probability density function given by

                             f (x) = k (2xx2),   for 0 £ x £ 2

                             f (x) = 0,                  elsewhere.

(a)     Find the value of k.

(b)     Find P(0.25 £ x £ 0.5).

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

53.     Ian and Karl have been chosen to represent their countries in the Olympic discus throw. Assume that the distance thrown by each athlete is normally distributed. The mean distance thrown by Ian in the past year was 60.33 m with a standard deviation of 1.95 m.

(a)     In the past year, 80% of Ian’s throws have been longer than x metres.
Find x, correct to two decimal places.

(3)

(b)     In the past year, 80% of Karl’s throws have been longer than 56.52 m. If the mean distance of his throws was 59.39 m, find the standard deviation of his throws, correct to two decimal places.

(3)

(c)     This year, Karl’s throws have a mean of 59.50 m and a standard deviation of 3.00 m. Ian’s throws still have a mean of 60.33 m and standard deviation 1.95 m. In a competition an athlete must have at least one throw of 65 m or more in the first round to qualify for the final round. Each athlete is allowed three throws in the first round.

(i)      Determine which of these two athletes is more likely to qualify for the final on their first throw.

(ii)     Find the probability that both athletes qualify for the final.

(11)

(Total 17 marks)

 

 

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

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

(3)

(b)     Find P(X £ 3).

(1)

          Let Y be another random variable, independent of X, with a Poisson distribution such that E(Y) = 2.

(c)     Find P(X + Y < 4).

(2)

(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.

(4)

(Total 10 marks)

 

 

55.     A discrete random variable X has its probability distribution given by

P(X = x) = k(x + 1), where x is 0, 1, 2, 3, 4.

(a)     Show that k =

(b)     Find E(X).

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

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

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

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

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

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

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

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

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

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

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

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

(Total 6 marks)

 


 

56.     The speeds of cars at a certain point on a straight road are normally distributed with mean μ and standard deviation σ. 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 μ and σ.

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

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

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(Total 6 marks)

 


 

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

(a)     P(3 £ X £ 5);

(b)     P(X ³ 3);

(c)     P(3 £ X < 5ïX ³ 3).

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

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

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

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

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

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

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

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

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

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

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

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

(Total 6 marks)

 


 

58.     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.

(2)

          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 , show that n = 6.

(4)

          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.

(4)

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

(3)

(Total 13 marks)

 

 

59.     The continuous random variable X has probability density function

          f (x) = x(1 + x2) for 0 £ x £ 2,
f (x) = 0 otherwise.

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

(2)

(b)     Write down the mode of X.

(1)

(c)     Find the mean of X.

(4)

(d)     Find the median of X.

(5)

(Total 12 marks)

 


 

60.     Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n Î +.

(Total 9 marks)

 

 

61.     When a fair die is thrown, the probability of obtaining a “6” is .

          Charles throws such a die repeatedly.

(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.

(10)

(b)     On which throw is he most likely to throw his first “6”?

(2)

(Total 12 marks)

 

 

62.     Consider the 10 data items x1, x2, ... x10. Given that  = 1341 and the standard deviation is 6.9, find the value of .

(Total 6 marks)

 

 

63.     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 probability that

(a)     only medical students are chosen;

(b)     all three law students are chosen.

(Total 6 marks)

 


 

64.     The probability density function f (x) of the continuous random variable X is defined on the interval [0, a] by

          Find the value of a.

(Total 6 marks)

 

 

65.     Given that (A È B)¢ = Æ, P(A¢|B) =  and P(A) = , find P(B).

(Total 6 marks)

 

 

66.     A company buys 44% of its stock of bolts from manufacturer A and the rest from manufacturer B. The diameters of the bolts produced by each manufacturer follow a normal distribution with a standard deviation of 0.16 mm.

          The mean diameter of the bolts produced by manufacturer A is 1.56 mm.
24.2% of the bolts produced by manufacturer B have a diameter less than 1.52 mm.

(a)     Find the mean diameter of the bolts produced by manufacturer B.

(3)

          A bolt is chosen at random from the company’s stock.

(b)     Show that the probability that the diameter is less than 1.52 mm is 0.312, to three significant figures.

(4)

(c)     The diameter of the bolt is found to be less than 1.52 mm. Find the probability that the bolt was produced by manufacturer B.

(3)

(d)     Manufacturer B makes 8000 bolts in one day. It makes a profit of $1.50 on each bolt sold, on condition that its diameter measures between 1.52 mm and 1.83 mm. Bolts whose diameters measure less than 1.52 mm must be discarded at a loss of $0.85 per bolt.
Bolts whose diameters measure over 1.83 mm are sold at a reduced profit of $0.50 per bolt.

          Find the expected profit for manufacturer B.

(6)

(Total 16 marks)

 

 

67.     Let X be a random variable with a Poisson distribution, such that P(X > 2) = 0.404.
Find P(X < 2).

(Total 4 marks)

 


 

68.     Box A contains 6 red balls and 2 green balls. Box B contains 4 red balls and 3 green balls. A fair cubical die with faces numbered 1, 2, 3, 4, 5, 6 is thrown. If an even number is obtained, a ball is selected from box A; if an odd number is obtained, a ball is selected from box B.

(a)     Calculate the probability that the ball selected was red.

(b)     Given that the ball selected was red, calculate the probability that it came from box B.

(Total 6 marks)

 

 

69.     A random variable X is normally distributed with mean m and variance s2. If P (X > 6.2) = 0.9474 and P (X < 9.8) = 0.6368, calculate the value of m and of s.

(Total 6 marks)

 


 

70.     There are 25 disks in a bag. Some of them are black and the rest are white. Two are simultaneously selected at random. Given that the probability of selecting two disks of the same colour is equal to the probability of selecting two disks of different colour, how many black disks are there in the bag?

(Total 6 marks)

 

 

71.     In a game a player pays an entrance fee of $n. He then selects one number from 1, 2, 3, 4, 5, 6 and rolls three standard dice.

If his chosen number appears on all three dice he wins four times his entrance fee.

If his number appears on exactly two of the dice he wins three times the entrance fee.

If his number appears on exactly one die he wins twice the entrance fee.

If his number does not appear on any of the dice he wins nothing.

(a)     Copy and complete the probability table below.

Profit ($)

n

n

2n

3n

Probability

 

 

 

(4)

(b)     Show that the player’s expected profit is $.

(2)

(c)     What should the entrance fee be so that the player’s expected loss per game is 34 cents?

(2)

(Total 8 marks)

 

 

72.     The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 200 g. The weights of 88% of the loaves are less than 220 g.

Find the standard deviation.

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(Total 6 marks)

 


 

73.     Let A and B be events such that P(A) = , P(B | A) =  and and P(A È B) = .

(a)     Find P(A Ç B).

(b)     Find P(B).

(c)     Show that A and B are not independent.

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(Total 6 marks)

 


 

74.     The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5.

(a)     Find the probability that more than two accidents will occur on a given Monday.

(b)     Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day.

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(Total 6 marks)

 


 

75.     The time, T minutes, required by candidates to answer a question in a mathematics examination has probability density function

(a)     Find

(i)      m, the expected value of T;

(ii)     s2, the variance of T.

(7)

(b)     A candidate is chosen at random. Find the probability that the time taken by this candidate to answer the question lies in the interval [ms, m].

(5)

(Total 12 marks)

 

 

76.     Andrew shoots 20 arrows at a target. He has a probability of 0.3 of hitting the target. All shots are independent of each other. Let X denote the number of arrows hitting the target.

(a)     Find the mean and standard deviation of X.

(5)

(b)     Find

(i)      P(X = 5);

(ii)     P(4 £ X £ 8).

(6)

          Bill also shoots arrows at a target, with probability of 0.3 of hitting the target. All shots are independent of each other.

(c)     Calculate the probability that Bill hits the target for the first time on his third shot.

(3)

(d)     Calculate the minimum number of shots required for the probability of at least one shot hitting the target to exceed 0.99.

(5)

(Total 19 marks)

 


 

77.     Buses arrive at a bus-stop T minutes apart, where T may be assumed to have an exponential distribution with probability density function

                    f (t) =

(a)     Show that

(i)      P(T > t) = ;

(ii)     P(T £ t + s | T > t) = , where s > 0.

(10)

 

(b)     Bill arrives at the bus-stop five minutes after the previous bus arrived at the bus-stop. Find the probability that the next bus arrives within 10 minutes of his arrival at the bus-stop.

(4)

(Total 14 marks)

 


 

78.     The random variable X is thought to have a geometric distribution with probability mass function

          P(X = x) = p(1 – p)x–1 for x Î +,

where p is an unknown parameter.

The value of X is recorded on 100 independent occasions with the following results.

x

Frequency

1

46

2

26

3

16

4

10

5 or more

3

(a)     (i)      Calculate the mean of these data.

(ii)     Deduce that the estimated value of p is .

(4)

(b)     Calculate an appropriate value of χ2. Test, at the 5% significance level, whether or not these data can be modelled by a geometric distribution.

(13)

(Total 17 marks)

 

 

79.     When Bill shoots an arrow at a target, he has a probability 0.6 of hitting the target. Each shot is independent of all other shots.

(a)     Find the probability of

(i)      hitting the target five times in eight shots;

(ii)     hitting the target for the fifth time on the eighth shot.

(6)

(b)     One morning, he decides to shoot arrows at the target and to stop as soon as he hits the target for the tenth time. Find the mean and standard deviation of the number of shots required.

(5)

(Total 11 marks)

 

 

80.     Bag 1 contains 4 red cubes and 5 blue cubes. Bag 2 contains 7 red cubes and 2 blue cubes.

Two cubes are drawn at random, the first from Bag 1 and the second from Bag 2.

(a)     Find the probability that the cubes are of the same colour.

(b)     Given that the cubes selected are of different colours, find the probability that the red cube was selected from Bag 1.

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(Total 6 marks)

 


 

81.     The random variable X follows a Poisson distribution. Given that P(X £ 1) = 0.2, find

(a)     the mean of the distribution;

(b)     P(X £ 2).

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(Total 6 marks)

 


 

82.     A certain type of vegetable has a weight which follows a normal distribution with mean 450 grams and a standard deviation 50 grams.

(a)     In a load of 2000 of these vegetables, calculate the expected number with a weight greater than 525 grams.

(b)     Find the upper quartile of the distribution.

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(Total 6 marks)

 


 

83.     A bag contains a very large number of ribbons. One quarter of the ribbons are yellow and the rest are blue. Ten ribbons are selected at random from the bag.

(a)     Find the expected number of yellow ribbons selected.

(2)

(b)     Find the probability that exactly six of these ribbons are yellow.

(2)

(c)     Find the probability that at least two of these ribbons are yellow.

(3)

(d)     Find the most likely number of yellow ribbons selected.

(4)

(e)     What assumption have you made about the probability of selecting a yellow ribbon?

(1)

(Total 12 marks)

 

 

84.     The continuous random variable X has probability density function

                      f (x) =

(a)     Find the exact value of k.

(5)

(b)     Find the mode of X.

(2)

(c)     Calculate P(1 £ X £ 2).

(3)

(Total 10 marks)

 


 

85.     A chocolate manufacturer puts gift vouchers at random into 15% of all chocolate bars produced. A customer must collect five vouchers to qualify for a gift.

(a)     Barry goes into a shop and buys 20 of these bars. Find the probability that he qualifies for a gift.

(3)

(b)     John goes into a shop and buys n of these bars. Find the smallest value of n for which the probability of qualifying for a gift exceeds .

(4)

(c)     Martina goes into a shop and buys these bars one at a time: she opens them to see if they contain a voucher. She obtains her 5th voucher on the Xth bar bought.

(i)      Write down an expression for P(X = x), valid for x ³ 5.

(ii)     Calculate E (X).

(iii)    Show that  = .

(iv)    Show that if P(X = x) > P(X = x −1) then x < . Deduce the most probable value of X.

(16)

(Total 23 marks)

 


 

86.     The events A and B are such that P(A) = 0.5, P(B) = 0.3, P(A È B) = 0.6.

(a)     (i)      Find the value of P(A Ç B).

(ii)     Hence show that A and B are not independent.

(b)     Find the value of P(B | A).

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(Total 6 marks)

 


 

87.     In an experiment, a trial is repeated n times. The trials are independent and the probability p of success in each trial is constant. Let X be the number of successes in the n trials. The mean of X is 0.4 and the standard deviation is 0.6.

(a)     Find p.

(b)     Find n.

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(Total 6 marks)

 


 

88.     A continuous random variable X has probability density function f defined by

                            f (x) =

Find the exact value of E(X).

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(Total 6 marks)

 


 

89.     A biology test consists of seven multiple choice questions. Each question has five possible answers, only one of which is correct. At least four correct answers are required to pass the test. Juan does not know the answer to any of the questions so, for each question, he selects the answer at random.

(a)     Find the probability that Juan answers exactly four questions correctly.

(b)     Find the probability that Juan passes the biology test.

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(Total 6 marks)

 


 

90.     A continuous random variable X has the probability density function f given by

                   f (x) =

(a)     State the mode of X.

(b)     Find the exact value of E (X).

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(Total 6 marks)

 


 

91.     The lengths of a particular species of lizard are normally distributed with a mean length of 50 cm and a standard deviation of 4 cm. A lizard is chosen at random.

(a)     Find the probability that its length is greater than 45 cm.

(b)     Given that its length is greater than 45 cm, find the probability that its length is greater than 55 cm.

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(Total 6 marks)

 


 

92.     The time, T minutes, spent each day by students in Amy’s school sending text messages may be modelled by a normal distribution.

          30% of the students spend less than 10 minutes per day.

          35% spend more than 15 minutes per day.

(a)     Find the mean and standard deviation of T.

(6)

          The number of text messages received by Amy during a fixed time interval may be modelled by a Poisson distribution with a mean of 6 messages per hour.

(b)     Find the probability that Amy will receive exactly 8 messages between 16:00 and 18:00 on a random day.

(3)

(c)     Given that Amy has received at least 10 messages between 16:00 and 18:00 on a random day, find the probability that she received 13 messages during that time.

(5)

(d)     During a 5-day week, find the probability that there are exactly 3 days when Amy receives no messages between 17:45 and 18:00.

(4)

(Total 18 marks)

 

 

93.     Twenty candidates sat an examination in French. The sum of their marks was 826 and the sum of the squares of their marks was 34132. Two candidates sat the examination late and their marks were a and b. The new mean and variance were calculated, giving the following results:

                    mean = 42 and variance = 32.

Find a set of possible values of a and b.

(Total 8 marks)

 

 

94.     (a)     Find the probability that a number, chosen at random between 200 and 800 inclusive, will be a multiple of 9.

(3)

(b)     Find the sum of the numbers between 200 and 800 inclusive, which are multiples of 6, but not multiples of 9.

(8)

(Total 11 marks)

 


 

95.     The times taken for buses travelling between two towns are normally distributed with a mean of 35 minutes and standard deviation of 7 minutes.

(a)     Find the probability that a randomly chosen bus completes the journey in less than 40 minutes.

(2)

(b)     90% of buses complete the journey in less than t minutes. Find the value of t.

(5)

(c)     A random sample of 10 buses had their travel time between the two towns recorded. Find the probability that exactly 6 of these buses complete the journey in less than 40 minutes.

(4)

(Total 11 marks)

 

 

96.     The number of bus accidents that occur in a given period of time has a Poisson distribution with a mean of 0.6 accidents per day.

(a)     Find the probability that at least two accidents occur on a randomly chosen day.

(4)

(b)     Find the most likely number of accidents occurring on a randomly chosen day.

Justify your answer.

(3)

(c)     Find the probability that no accidents occur during a randomly chosen seven-day week.

(3)

(Total 10 marks)

 


 

97.     A zoologist believes that the number of eggs laid in the Spring by female birds of a certain breed follows a Poisson law. She observes 100 birds during this period and she produces the following table.

Number of eggs laid

Frequency

0

10

1

19

2

34

3

23

4

10

5

4

(a)     Calculate the mean number of eggs laid by these birds.

(2)

(b)     The zoologist wishes to determine whether or not a Poisson law provides a suitable model.

(i)      Write down appropriate hypotheses.

(ii)     Carry out a test at the 1% significance level, and state your conclusion.

(16)

(Total 18 marks)

 

 

98.     Let X1, X2, ....., X20 be independent random variables each having a geometric distribution with probability of success p equal to 0.6.

Let Y =

(a)     Explain why the random variable Y has a negative binomial distribution.

(2)

(b)     Find the mean and variance of Y.

(4)

(c)     Calculate P(Y = 30).

(4)

(Total 10 marks)

 


 

99.     (a)     The random variable X has a geometric distribution with parameter p =

What is the value of P(X £ 4)?

(3)

(b)     A magazine publisher promotes his magazine by putting a concert ticket at random in one out of every four magazines. If you need 8 tickets to take friends to the concert, what is the probability that you will find your last ticket when you buy the 20th magazine?

(3)

(c)     How are the two distributions in parts (a) and (b) related?

(2)

(Total 8 marks)

 

 

100.   (a)     If Y has a Poisson distribution Po (m), show that P(Y = y + 1) =  P(Y = y).

(3)

(b)     The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.

Number of cars

0

1

2

3

4

5

Frequency

4

18

19

20

11

8

          Carry out a χ2 goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.

(11)

(Total 14 marks)

 


 

101.   A furniture manufacturer makes tables. A table leg is considered to be oversize if its width is greater than 10.5 cm and undersize if its width is less than 9.5 cm. From past experience it is found that 2% of the table legs that are made are oversize and that 4% of the table legs are undersize. The widths of the table legs are normally distributed with mean m cm and standard deviation s cm. Find the value of m and of s.

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(Total 6 marks)

 


 

102.   In a promotion to try to increase the sales of a particular brand of breakfast cereal, a picture of a soccer player is put in each packet. There are ten different pictures available. Each picture is equally likely to be found in any packet of breakfast cereal.

Charlotte buys four packets of breakfast cereal.

(a)     Find the probability that the four pictures in these packets are all different.

(2)

(b)     Of the ten players whose pictures are in the packets, her favourites are Alan and Bob. Find the probability that she finds at least one picture of a favourite player in these four packets.

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(4)

(Total 6 marks)

 


 

103.   On a particular road, serious accidents occur at an average rate of two per week and can be modelled using a Poisson distribution.

(a)     (i)      What is the probability of at least eight serious accidents occurring during a particular four-week period?

(ii)     Assume that a year consists of thirteen periods of four weeks. Find the probability that in a particular year, there are more than nine four-week periods in which at least eight serious accidents occur.

(10)

(b)     Given that the probability of at least one serious accident occurring in a period of n weeks is greater than 0.99, find the least possible value of n, nÎ +.

(8)

(Total 18 marks)

 

 

104.   A continuous random variable X has probability density function defined by

                            f (x) =

(a)     Find the exact value of the constant c in terms of p.

(5)

(b)     Sketch the graph of f (x) and hence state the mode of the distribution.

(3)

(c)     Find the exact value of E(X).

(4)

(Total 12 marks)

 


 

105.   Juan plays a quiz game. The scores he achieves on the separate topics may be modelled by independent normal distributions.

(a)     On the topic of sport, the scores have the distribution N (75, 122).

Find the probability that Juan scores less than 57 points on the topic of sport.

(2)

(b)     On the topic of literature, Juan’s scores have a mean of 45, and 30% of his scores are greater than 50.

Find the standard deviation of his scores on the topic of literature.

(3)

(c)     Juan claims that he scores better in current affairs than in sport. He achieves the following scores on current affairs in 10 separate quizzes.

                       91 84 75 92 88 71 83 90 85 78

          Perform a hypothesis test at the 5% significance level to decide whether there is evidence to support his claim.

(6)

(Total 11 marks)

 


 

106.   (a)     The exponential distribution has the probability density function

                    f (x) =

Show that the mean is greater than the median.

(4)

(b)     The time in seconds between arrivals of butterflies on a flowering bush can be modelled by an exponential distribution with parameter l = 0.1.

A butterfly arrives on the bush.

(i)      Calculate the probability that no other butterfly arrives within 20 seconds.

(ii)     Given that no other butterfly has arrived within 20 seconds, calculate the probability that the next butterfly arrives within 50 seconds of the first.

(6)

(c)     The number of goals scored by a soccer team in a period of duration t minutes follows a Poisson distribution with mean

(i)      Write down the probability that no goals are scored during a period of duration t minutes.

(ii)     The random variable T is defined as the length of time, in minutes, between successive goals. Show that T follows an exponential distribution.

(5)

(Total 15 marks)

 


 

107.   The random variable T has the probability density function

                               f (t) =

Find

(a)     P(T = 0);

(2)

(b)     the interquartile range.

(5)

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(Total 7 marks)

 


 

108.   The probability distribution of a discrete random variable X is defined by

          P(X = x) = cx(5 − x), x = 1, 2, 3, 4.

(a)     Find the value of c.

(3)

(b)     Find E(X).

(3)

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(Total 6 marks)

 


 

109.   Let A and B be events such that P(A) = 0.6, P(A È B) = 0.8 and P(A | B) = 0.6.

Find P(B).

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(Total 6 marks)

 


 

110.   A company produces computer microchips, which have a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.7 months.

(a)     If a microchip is guaranteed for 84 months find the probability that it will fail before the guarantee ends.

(2)

(b)     The probability that a microchip does not fail before the end of the guarantee is required to be 99%. For how many months should it be guaranteed?

(2)

(c)     A rival company produces microchips where the probablity that they will fail after 84 months is 0.88. Given that the life expectancy also follows a normal distribution with standard deviation 3.7 months, find the mean.

(2)

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(Total 6 marks)

 


 

111.   Only two international airlines fly daily into an airport. UN Air has 70 flights a day and IS Air has 65 flights a day. Passengers flying with UN Air have an 18% probability of losing their luggage and passengers flying with IS Air have a 23% probability of losing their luggage. You overhear someone in the airport complain about her luggage being lost.

Find the probability that she travelled with IS Air.

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(Total 6 marks)

 


 

112.   The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at any given time are independent of one another and can be modelled using a Poisson Distribution with mean 0.2 per day.

(a)     Determine the probability that there will be exactly four breakdowns during the month of June (June has 30 days).

(3)

(b)     Determine the probability that there are more than 3 breakdowns during the month of June.

(2)

(c)     Determine the probability that there are no breakdowns during the first five days of June.

(2)

(d)     Find the probability that the first breakdown in June occurs on June 3rd.

(3)

(e)     It costs 1850 Euros to service the lifts when they have breakdowns. Find the expected cost of servicing lifts for the month of June.

(1)

(f)      Determine the probability that there will be no breakdowns in exactly 4 out of the first 5 days in June.

(2)

(Total 13 marks)

 


 

113.   A continuous random variable X has probability density function

                    f (x) =

Find the probability that X lies between the mean and the mode.

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(Total 6 marks)

 


 

114.   Over a one month period, Ava and Sven play a total of n games of tennis.

          The probability that Ava wins any game is 0.4. The result of each game played is independent of any other game played.

Let X denote the number of games won by Ava over a one month period.

(a)     Find an expression for P(X = 2) in terms of n.

(3)

(b)     If the probability that Ava wins two games is 0.121 correct to three decimal places, find the value of n.

(3)

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(Total 6 marks)

 


 

115.   The distance travelled by students to attend Gauss College is modelled by a normal distribution with mean 6 km and standard deviation 1.5 km.

(a)     (i)      Find the probability that the distance travelled to Gauss College by a randomly selected student is between 4.8 km and 7.5 km.

(ii)     15% of students travel less than d km to attend Gauss College. Find the value of d.

(7)

          At Euler College, the distance travelled by students to attend their school is modelled by a normal distribution with mean m km and standard deviation s km.

(b)     If 10% of students travel more than 8 km and 5% of students travel less than 2 km, find the value of m and of s.

(6)

          The number of telephone calls, T, received by Euler College each minute can be modelled by a Poisson distribution with a mean of 3.5.

(c)     (i)      Find the probability that at least three telephone calls are received by Euler College in each of two successive one-minute intervals.

(ii)     Find the probability that Euler College receives 15 telephone calls during a randomly selected five-minute interval.

(8)

(Total 21 marks)

 

 

116.   Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When she tosses it, the score is defined as the number on the uppermost face. One day, she decides to toss the die repeatedly until all the possible scores have occurred at least once.

(a)     Having thrown the die once, she lets X2 denote the number of additional throws required to obtain a different number from the one obtained on the first throw. State the distribution of X2 and hence find E(X2).

(3)

(b)     She then lets X3 denotethe number of additional throws required to obtain a different number from the two numbers already obtained. State the distribution of X3 and hence find E(X3).

(2)

(c)     By continuing the process, show that the expected number of tosses needed to obtain all six possible scores is 14.7.

(5)

(Total 10 marks)

 


 

117.   A factory makes wine glasses. The manager claims that on average 2% of the glasses are imperfect. A random sample of 200 glasses is taken and 8 of these are found to be imperfect.

Test the manager’s claim at a 1% level of significance using a one-tailed test.

(Total 7 marks)

 

 

118.   The number of telephone calls received by a helpline over 80 one-minute periods are summarized in the table below.

Number of calls

0

1

2

3

4

5

6

Frequency

9

12

22

10

11

8

8

(a)     Find the exact value of the mean of this distribution.

(2)

(b)     Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.

(12)

(Total 14 marks)

 


 

119.   The heights, x metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.

                

(a)     Calculate unbiased estimates of the population mean and the population variance.

(3)

(b)     The Head of Mathematics decided to use a χ2 test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.

Interval

x < 1.60

1.60 £ x < 1.65

1.65 £ x < 1.70

1.70 £ x < 1.75

1.75 £ x < 1.80

x ³ 1.80

Frequency

5

34

70

72

48

12

(i)      State suitable hypotheses.

(ii)     Calculate the value of the χ2 statistic and state your conclusion using a 10% level of significance.

(12)

(Total 15 marks)

 

 

120.   (a)     The independent random variables X and Y have Poisson distributions and Z = X +Y. The means of X and Y are l and m respectively. By using the identity

                   

show that Z has a Poisson distribution with mean (l +m).

(6)

(b)     Given that U1, U2, U3, … are independent Poisson random variables each having mean m, use mathematical induction together with the result in (a) to show that  has a Poisson distribution with mean nm.

(6)

(Total 12 marks)

 

 

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