5.02b Expectation and variance: discrete random variables

2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.

1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
2. Find the probability distribution of \(X\).
3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
4. Find the probability that Linda scores more points in round 2 than in round 1.

4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2 \\ \alpha , & x = - 1,0 \\ 0.1 , & x = 1,2 . \end{array}$$ Find

1. \(\alpha\),
2. \(\mathrm { P } ( - 1 \leq X < 2 )\),
3. \(\mathrm { F } ( 0.6 )\),
4. the value of \(a\) such that \(\mathrm { E } ( a X + 3 ) = 1.2\),
5. \(\operatorname { Var } ( X )\),
6. \(\operatorname { Var } ( 3 X - 2 )\).

2. A farmer sells boxes of eggs. The eggs are sold in boxes of 6 eggs and boxes of 12 eggs in the ratio \(n : 1\) A random sample of three boxes is taken.
The number of eggs in the first box is denoted by \(X _ { 1 }\) The number of eggs in the second box is denoted by \(X _ { 2 }\) The number of eggs in the third box is denoted by \(X _ { 3 }\) The random variable \(T = X _ { 1 } + X _ { 2 } + X _ { 3 }\) Given that \(\mathrm { P } ( T = 18 ) = 0.729\)

1. show that \(n = 9\)
2. find the sampling distribution of \(T\) The random variable \(R\) is the range of \(X _ { 1 } , X _ { 2 } , X _ { 3 }\)
3. Using your answer to part (b), or otherwise, find the sampling distribution of \(R\)

6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
2. Suggest a suitable sampling frame.
3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
   When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)

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A bag contains a large number of coins. It only contains 20 p and 50 p coins. A random sample of 3 coins is taken from the bag.

1. List all the possible combinations of 3 coins that might be taken. Let \(\bar { X }\) represent the mean value of the 3 coins taken.
   Part of the sampling distribution of \(\bar { X }\) is given below.

   \(\bar { x }\)	20	\(a\)	\(b\)	50
   \(\mathrm { P } ( \bar { X } = \bar { x } )\)	\(\frac { 4913 } { 8000 }\)	\(c\)	\(d\)	\(\frac { 27 } { 8000 }\)

2. Write down the value of \(a\) and the value of \(b\) The probability of taking a 20p coin at random from the bag is \(p\) The probability of taking a 50p coin at random from the bag is \(q\)
3. Find the value of \(p\) and the value of \(q\)
4. Hence, find the value of \(c\) and the value of \(d\) Let \(M\) represent the mode of the 3 coins taken at random from the bag.
5. Find the sampling distribution of \(M\)

1. A bag contains a large number of counters with an odd number or an even number written on each.

Odd and even numbered counters occur in the ratio \(4 : 1\) In a game a player takes a random sample of 4 counters from the bag.
The player scores
5 points for each counter taken that has an even number written on it
2 points for each counter taken that has an odd number written on it
The random variable \(X\) represents the total score, in points, from the 4 counters.

1. Find the sampling distribution of \(X\) A random sample of \(n\) sets of 4 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a total score of exactly 14
2. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.95\)

5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$

1. List all the possible values of \(Y\).
2. Find the sampling distribution of \(Y\).
3. Find \(\mathrm { E } ( Y )\).

6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
   The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.

2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)

1. 1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
   2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)

5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below

1. List all the possible samples of two coins which may be selected.
2. Find the value of \(q\), the value of \(r\) and the value of \(s\)

7. Last year \(4 \%\) of cars tested in a large chain of garages failed an emissions test. A random sample of \(n\) of these cars is taken. The number of cars that fail the test is represented by \(X\) Given that the standard deviation of \(X\) is 1.44

1. 1. find the value of \(n\)
   2. find \(\mathrm { E } ( X )\) A random sample of 20 of the cars tested is taken.
2. Find the probability that all of these cars passed the emissions test. Given that at least 1 of these cars failed the emissions test,
3. find the probability that exactly 3 of these cars failed the emissions test. A car mechanic claims that more than \(4 \%\) of the cars tested at the garage chain this year are failing the emissions test. A random sample of 125 of these cars is taken and 10 of these cars fail the emissions test.
4. Using a suitable approximation, test whether or not there is evidence to support the mechanic's claim. Use a \(5 \%\) level of significance and state your hypotheses clearly.
   

2. Crispy-crisps produces packets of crisps. During a promotion, a prize is placed in \(25 \%\) of the packets. No more than 1 prize is placed in any packet. A box contains 6 packets of crisps.

1. 1. Write down a suitable distribution to model the number of prizes found in a box.
   2. Write down one assumption required for the model.
2. Find the probability that in 2 randomly selected boxes, only 1 box contains exactly 1 prize.
3. Find the probability that a randomly selected box contains at least 2 prizes. Neha buys 80 boxes of crisps.
4. Using a normal approximation, find the probability that no more than 30 of the boxes contain at least 2 prizes.

6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match. Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets. A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\) respectively. The random variable \(M\) represents the maximum value of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\)

1. List all the samples where \(M \neq 5\)
2. Find the sampling distribution of \(M\)
3. Write down the mode of \(S _ { 1 }\) and the mode of \(M\)

1. A salesman sells insurance to people. Each day he chooses a number of people to contact. The probability that the salesman sells insurance to a person he contacts is 0.05

On Monday he chooses to contact 10 people.

1. Find the probability that on Monday the salesman sells insurance to
   1. exactly 1 person,
   2. at least 3 people.
2. Find the number of people he should contact each day in order to sell insurance, on average, to 3 people per day.
3. Calculate the least number of people he must choose to contact on Friday, so that the probability of selling insurance to at least 1 person on Friday exceeds 0.99
   

Spany sells seeds and claims that \(5 \%\) of its pansy seeds do not germinate. A packet of pansy seeds contains 20 seeds. Each seed germinates independently of the other seeds.

1. Find the probability that in a packet of Spany's pansy seeds
   1. more than 2 but fewer than 5 seeds do not germinate,
   2. more than 18 seeds germinate. Jem buys 5 packets of Spany's pansy seeds.
2. Calculate the probability that all of these packets contain more than 18 seeds that germinate. Jem believes that Spany's claim is incorrect. She believes that the percentage of pansy seeds that do not germinate is greater than 5\%
3. Write down the hypotheses for a suitable test to examine Jem's belief. Jem planted all of the 100 seeds she bought from Spany and found that 8 did not germinate.
4. Using a suitable approximation, carry out the test using a \(5 \%\) level of significance.

1. The independent random variables \(W\) and \(X\) have the following distributions.

$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$

1. Write down the value of the variance of \(W\)
2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)

1. A bag contains 10 counters each with exactly one number written on it.

There are 6 counters with the number 7 on them
There are 3 counters with the number 8 on them
There is 1 counter with the number 9 on it
A random sample of 3 counters is taken from the bag (without replacement).
These counters are then put back in the bag.
This process is then repeated until 20 samples have been taken.
The random variable \(Y\) represents the number of these 20 samples that contain the counter with the number 9 on it.

1. 1. Find the mean of \(Y\)
   2. Find the variance of \(Y\) A random sample of 3 counters is chosen from the bag (without replacement).
2. List all possible samples where the median of the numbers on the 3 counters is 7
3. Find the sampling distribution of the median of the numbers on the 3 counters.

1. In a large population \(40 \%\) of adults use online banking.

A random sample of 50 adults is taken.
The random variable \(X\) represents the number of adults in the sample that use online banking.

1. Find
   1. \(\mathrm { P } ( X = 26 )\)
   2. \(\mathrm { P } ( X \geqslant 26 )\)
   3. the smallest value of \(k\) such that \(\mathrm { P } ( X \leqslant k ) > 0.4\) A random sample of 600 adults is taken.
2. 1. Find, using a normal approximation, the probability that no more than 222 of these 600 adults use online banking.
   2. Explain why a normal approximation is suitable in part (b)(i)

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a \(10 \%\) level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games,
2. fewer than half of the games. Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.
3. Calculate the mean and variance for the number of these 60 games that Bhim loses.
4. Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

6. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac { 1 } { 4 }\). The probability of rejection in either tail should be as close as possible to 0.025
3. Find the actual significance level of this test. In the sample of 50 the actual number of faulty bolts was 8 .
4. Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
5. Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.

3. An airline knows that overall \(3 \%\) of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.

1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. By using a suitable approximation, find the probability that
2. more than 196 passengers turn up for this flight,
3. there is at least one empty seat on this flight.

6. The owner of a small restaurant decides to change the menu. A trade magazine claims that \(40 \%\) of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners \(X\) who choose organic foods.
2. Find \(\mathrm { P } ( 5 < X < 15 )\).
3. Find the mean and standard deviation of \(X\). The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
4. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.
   (5)

5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 . Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one,
2. more than three. A customer bought three boxes.
3. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
4. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
5. Find the probability that a randomly chosen egg weighs more than 68 g .

6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.

1. Find the mean and variance of the value of the coins. A random sample of 2 coins is chosen from the bag.
2. List all the possible samples that can be drawn.
3. Find the sampling distribution of the mean value of these samples.