2.04b Binomial distribution: as model B(n,p)

2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are \(0.45,0.25\) and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.

1. On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
2. On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
   1. at most 12 of these guests require a continental breakfast;
   2. more than 10 but fewer than 20 of these guests require no breakfast.
3. When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.

6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.

1. Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
   1. on at most 7 nights;
   2. on at least 11 nights;
   3. on more than 5 nights but fewer than 10 nights.
2. Calculate the probability that, during a period of \(\mathbf { 3 }\) weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
3. Barry claims that, during the summer, the number of nights per week, \(S\), on which Sylvester chooses to stay outside can be modelled by a binomial distribution with \(n = 7\) and \(p = \frac { 5 } { 7 }\).
   1. Assuming that Barry's claim is correct, find the mean and the variance of \(S\).
   2. For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 . Comment on Barry's claim.
      (2 marks)

5

1. At a particular checkout in a supermarket, the probability that the barcode reader fails to read the barcode first time on any item is 0.07 , and is independent from item to item.
   1. Calculate the probability that, from a shopping trolley containing 17 items, the reader fails to read the barcode first time on exactly 2 of the items.
   2. Determine the probability that, from a shopping trolley containing 50 items, the reader fails to read the barcode first time on at most 5 of the items.
2. At another checkout in the supermarket, the probability that a faulty barcode reader fails to read the barcode first time on any item is 0.55 , and is independent from item to item. Determine the probability that, from a shopping trolley containing 50 items, this reader fails to read the barcode first time on at least 30 of the items.
3. At a third checkout in the supermarket, a record is kept of \(X\), the number of times per 50 items that the barcode reader fails to read a barcode first time. An analysis of the records gives a mean of 10 and a standard deviation of 6.8.
   1. Estimate \(p\), the probability that the barcode reader fails to read a barcode first time.
   2. Using your estimate of \(p\) and assuming that \(X\) can be modelled by a binomial distribution, estimate the standard deviation of \(X\).
   3. Hence comment on the assumption that \(X\) can be modelled by a binomial distribution.

5 Kirk and Les regularly play each other at darts.

1. The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game. Find the probability that, in a match of 15 games, Kirk wins:
   1. exactly 5 games;
   2. fewer than half of the games;
   3. more than 2 but fewer than 7 games.
2. Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
   1. Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
   2. To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results: $$\begin{array} { l l l l l l l l l l } 8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5 \end{array}$$ Calculate the mean and standard deviation of these values.
   3. Hence comment on the validity of Kirk's claim.

6 Customers at a supermarket can pay at a checkout either by cash, debit card or credit card.

1. The probability that a customer pays by cash is 0.22 . Calculate the probability that exactly 2 customers from a random sample of 24 customers pay by cash.
2. The probability that a customer pays by debit card is 0.45 . Determine the probability that the number of customers who pay by debit card from a random sample of \(\mathbf { 4 0 }\) customers is:
   1. fewer than 20 ;
   2. more than 15 ;
   3. at least 12 but at most 24 .
3. The random variable \(W\) denotes the number of customers who pay by credit card from a random sample of \(\mathbf { 2 0 0 }\) customers. Calculate values for the mean and the variance of \(W\).
   [0pt] [3 marks]

3 A ferry sails once each day from port D to port A. The ferry departs from D on time or late but never early. However, the ferry can arrive at A early, on time or late. The probabilities for some combined events of departing from \(D\) and arriving at \(A\) are shown in the table below.

1. Complete the table.
2. Write down the probability that, on a particular day, the ferry:
   1. both departs and arrives on time;
   2. departs late.
3. Find the probability that, on a particular day, the ferry:
   1. arrives late, given that it departed late;
   2. does not arrive late, given that it departed on time.
4. On three particular days, the ferry departs from port D on time. Find the probability that, on these three days, the ferry arrives at port A early once, on time once and late once. Give your answer to three decimal places.
   [0pt] [4 marks]
   1. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Answer space for question 3}

      \multirow{2}{*}{}		Arrive at A
      		Early	On time	Late	Total
      \multirow{2}{*}{Depart from D}	On time	0.16	0.56	0.08
      	Late
      	Total	0.22	0.65		1.00

      \end{table}

6

1. In a particular country, 35 per cent of the population is estimated to have at least one mobile phone. A sample of 40 people is selected from the population.
   Use the distribution \(\mathrm { B } ( 40,0.35 )\) to estimate the probability that the number of people in the sample that have at least one mobile phone is:
   1. at most 15 ;
   2. more than 10 ;
   3. more than 12 but fewer than 18 ;
   4. exactly equal to the mean of the distribution.
2. In the same country, 70 per cent of households have a landline telephone connection. A sample of 50 households is selected from all households in the country.
   Stating a necessary condition regarding this selection, estimate the probability that fewer than 30 households have a landline telephone connection.
   [0pt] [4 marks]

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p\). 16 shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   1. at least 8,
   2. between 4 and 9 inclusive.
   3. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 . \section*{June 2005}

6

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove that \(\mathrm { E } ( X ) = n p\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }\), show that \(\operatorname { Var } ( X ) = n p ( 1 - p )\).
   3. Given that \(X\) is found to have a mean of 3 and a variance of 2.97, find values for \(n\) and \(p\).
   4. Hence use a distributional approximation to estimate \(\mathrm { P } ( X > 2 )\).
2. Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.

14 The random variable \(T\) follows a binomial distribution where $$T \sim \mathrm {~B} ( 16,0.3 )$$ The mean of \(T\) is denoted by \(\mu\).
14

1. \(\quad\) Find \(\mathrm { P } ( T \leq \mu )\).
   14
2. Find the variance of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-19_2488_1716_219_153}

14 Yingtai visits her local gym regularly. After each visit she chooses one item to eat from the gym's cafe.
This could be an apple, a banana or a piece of cake.
She chooses the item independently each time.
The probability that Yingtai chooses each of these items on any visit is given by: $$\begin{aligned} \mathrm { P } ( \text { Apple } ) & = 0.2 \\ \mathrm { P } ( \text { Banana } ) & = 0.35 \\ \mathrm { P } ( \text { Cake } ) & = 0.45 \end{aligned}$$ For any four randomly selected visits to the gym, find the probability that Yingtai chose: 14

1. at least one banana.
   [0pt] [2 marks]
   14
2. the same item each time.
   

   14	apple twice and cake twice

16 It is believed that a coin is biased so that the probability of obtaining a head when the coin is tossed is 0.7 16

1. Assume that the probability of obtaining a head when the coin is tossed is indeed 0.7
   16
   1. (i) Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.
      16
   2. (ii) Find the mean number of heads obtained from 7 tosses of the coin.
      16
   3. Harry believes that the probability of obtaining a head for this coin is actually greater than 0.7 To test this belief he tosses the coin 35 times and obtains 28 heads. Carry out a hypothesis test at the \(10 \%\) significance level to investigate Harry's belief. \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-24_2492_1721_217_150}
      \includegraphics[max width=\textwidth, alt={}]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-28_2498_1722_213_147}

Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\) Naasir and his friends play the game 15 times.

1. Find the probability that Naasir wins
   1. exactly 2 games,
   2. more than 5 games. Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.

5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)

5.

1. The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
2. Write down the hypotheses that should be used to test the manager's suspicion.
3. Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
4. Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
5. Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
6. Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.

2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)

1. Find \(\mathrm { P } ( 5 \leq X < 12 )\). Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
2. Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
3. State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test. \section*{(Total for Question 2 is 8 marks)}

1. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.

Peta models \(H\) as \(\mathrm { B } ( 10,0.1 )\)

1. State two assumptions Peta needs to make to use her model.
2. Using Peta's model, find \(\mathrm { P } ( H \geqslant 4 )\) For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
3. find \(\mathrm { P } ( F = 5 )\) Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm { P } ( F = n )\) as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where \(\alpha\) is a constant.
4. Find the value of \(\alpha\)
5. Using Thomas' model, find \(\mathrm { P } ( F = 5 )\)
6. Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.

A company sells seeds and claims that \(55 \%\) of its pea seeds germinate.

1. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
2. Assuming that the company's claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
3. Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution. A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
4. Assuming that the company's claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
5. Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of \(55 \%\)

3. For a particular type of bulb, \(36 \%\) grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40 Russell selects a random sample of 5 bags of bulbs.

1. Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.
   (4) Maggie takes a random sample of \(n\) bulbs.
   Using a normal approximation, the probability that more than 244 of these \(n\) bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
2. Find the value of \(n\).
   (6)
   (Total 10 marks)

4 A fair six-sided dice is rolled repeatedly. Find the probability of the following events.

1. A five occurs for the first time on the fourth roll.
2. A five occurs at least once in the first four rolls.
3. A five occurs for the second time on the third roll.
4. At least two fives occur in the first three rolls. The dice is rolled repeatedly until a five occurs for the second time.
5. Find the expected number of rolls required for two fives to occur. Justify your answer.

1 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4 . Find the mean of the number of spins required.
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive.
3. Each pupil in a class of 30 spins the spinner until it shows the number 4 . Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\).

4 In one department of a firm, four employees are selected for promotion from a staff of eighteen.

1. In how many ways can four employees be selected? It is known that throughout the firm 5\% of those selected for promotion decline it.
2. If 100 employees are randomly selected for promotion in the firm, calculate the number expected to decline promotion.
3. If 20 employees are selected at random for promotion, use the binomial distribution to find the probability that fewer than five employees will decline promotion.

4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.

1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.

5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\), where \(p > 0.5\). A random sample of \(4 n\) observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean. It is given that \(\mathrm { E } ( \bar { X } ) = 180\) and \(\operatorname { Var } ( \bar { X } ) = 0.0225\).

1. Find
   1. the values of \(p\) and \(n\),
   2. \(\mathrm { P } ( \bar { X } < 179.8 )\),
   3. the value of \(a\) for which \(\mathrm { P } ( 180 - a < \bar { X } < 180 + a ) = 0.99\), giving your answer correct to 2 decimal places.
   4. State how you have used the Central Limit Theorem in part (i).

6

1. Verify that \(\left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + t ^ { 3 } + t ^ { 4 } + t ^ { 5 } \right)\).
2. An unbiased six-faced die is rolled \(r\) times. Show that the probability generating function for the total score is $$\left[ \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) } \right] ^ { r }$$
3. Hence show that the probability of the total score being ( \(r + 3\) ) is $$\left( \frac { 1 } { 6 } \right) ^ { r + 1 } r ( r + 1 ) ( r + 2 )$$