2.04c Calculate binomial probabilities

6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find

1. the probability that at least two of the apples in it are blemished,
2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
3. the probability that at most one tray contains at least two blemished apples,
4. the expected number of trays containing at least two blemished apples.
5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.

6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was \(35 \%\), and the percentage who voted for the Monster Raving Loony party was \(2 \%\).

1. Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
2. the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
3. the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
4. One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.

7

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
   2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
3. The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
   (3 marks)

7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .

1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
   (6 marks)
2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
   1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
   2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)

7.

1. Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
2. Write down the distribution of the number of sixes obtained when the ten dice are thrown.
3. Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
4. Find the probability that the mean number of sixes obtained is more than 1.8

5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)

1. Find the size of the test.
2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
   (d) On the axes below draw the graph of the power function for the statistician's test.
   (e) Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4 \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}

4. A poultry farm produces eggs which are sold in boxes of 6 . The farmer believes that the proportion, \(p\), of eggs that are cracked when they are packed in the boxes is approximately 5\%. She decides to test the hypotheses $$\mathrm { H } _ { 0 } : p = 0.05 \text { against } \mathrm { H } _ { 1 } : p > 0.05$$ To test these hypotheses she randomly selects a box of eggs and rejects \(\mathrm { H } _ { 0 }\) if the box contains 2 or more eggs that are cracked. If the box contains 1 egg that is cracked, she randomly selects a second box of eggs and rejects \(\mathrm { H } _ { 0 }\) if it contains at least 1 egg that is cracked. If the first or the second box contains no cracked eggs, \(\mathrm { H } _ { 0 }\) is immediately accepted and no further boxes are sampled.

1. Show that the power function of this test is $$1 - ( 1 - p ) ^ { 6 } - 6 p ( 1 - p ) ^ { 11 }$$
2. Calculate the size of this test. Given that \(p = 0.1\)
3. find the expected number of eggs inspected each time this test is carried out, giving your answer correct to 3 significant figures,
4. calculate the probability of a Type II error. Given that \(p = 0.1\) is an unacceptably high value for the farmer,
5. use your answer from part (d) to comment on the farmer's test.

3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.

1. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar.
2. Calculate the probability of a Type I error for this test. Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
3. calculate the probability of a Type II error for this test.

1. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5

A shopkeeper does not believe the manufacturer's claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are fewer than 3 long screws. In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm { H } _ { 0 }\) is rejected, otherwise it is accepted.

1. Find the size of test \(A\).
2. Find the size of test \(B\).
3. Find an expression for the power function of test \(B\) in terms of \(p\). Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.

   \(p\)	0.1	0.2	0.3	0.4
   Power test \(A\)	0.93	\(r\)	0.38	0.17
   Power test \(B\)	0.83	0.55	0.31	0.15

4. Find the value of \(r\). The shopkeeper believes that the value of \(p\) is less than 0.4
5. Suggest which of the tests the shopkeeper should use. Give a reason for your answer.

3 A child is trying to draw court cards from an ordinary pack of 52 cards (court cards are Kings, Queens and Jacks; there are 12 in a pack). She draws cards, one at a time, with replacement, from the pack. Find the probabilities of the following events.

1. She draws a court card for the first time on the sixth try.
2. She draws a court card at least once in the first six tries.
3. She draws a court card for the second time on the sixth try.
4. She draws at least two court cards in the first six tries.

2 A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.

1. Determine \(\mathrm { P } ( X = 4 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
   • The player has \(n\) attempts to score a goal.
     1. Determine the least value of \(n\) for which the probability that the player first scores a goal on the \(n\)th attempt is less than 0.001 .
     2. Determine the least value of \(n\) for which the probability that the player scores at least one goal in \(n\) attempts is at least 0.999.

5 A fair spinner has five faces, labelled 0, 1, 2, 3, 4.

1. State the distribution of the score when the spinner is spun once.
2. Determine the probability that, when the spinner is spun twice, one of the scores is less than 2 and the other is at least 2.
3. Find the variance of the total score when the spinner is spun 5 times.

4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find

5 In a recent report, it was stated that \(40 \%\) of working people have a degree. For the whole of this question, you should assume that this is true. A researcher wishes to interview a working person who has a degree. He asks working people at random whether they have a degree and counts the number of people he has to ask until he finds one with a degree.

1. Find the probability that he has to ask 5 people.
2. Find the mean number of people the researcher has to ask. Subsequently, the researcher decides to take a random sample from the population of working people.
3. A random sample of 5 working people is chosen. What is the probability that at least one of them has a degree?
4. How large a random sample of working people would the researcher need to take to ensure that the probability that at least one person has a degree is 0.99 or more?

2 A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.

1. Find the probability that the fifth person the researcher asks is the first to have watched the programme.
2. Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
3. Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
4. Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.

1. A chocolate manufacturer places special tokens in \(2 \%\) of the bars it produces so that each bar contains at most one token. Anyone who collects 3 of these tokens can claim a prize.

Andreia buys a box of 40 bars of the chocolate.

1. Find the probability that Andreia can claim a prize. Barney intends to buy bars of the chocolate, one at a time, until he can claim a prize.
2. Find the probability that Barney can claim a prize when he buys his 40th bar of chocolate.
3. Find the expected number of bars that Barney must buy to claim a prize.

1. A factory produces pins.

An engineer selects 40 independent random samples of 6 pins produced at the factory and records the number of defective pins in each sample.

1. Show that the proportion of defective pins in the 40 samples is 0.15 The engineer suggests that the number of defective pins in a sample of 6 can be modelled using a binomial distribution. Using the information from the sample above, a test is to be carried out at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. The value of the test statistic for this test is 2.689
2. Justifying the degrees of freedom used, carry out the test, at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. State your hypotheses clearly. The engineer later discovers that the previously recorded information was incorrect. The data should have been as follows.

   Number of defective pins	0	1	2	3	4	5	6
   Observed frequency	19	11	6	3	1	0	0

3. Describe the effect this would have on the value of the test statistic that should be used for the hypothesis test.
   Give reasons for your answer.

1. A researcher is investigating the number of female cubs present in litters of size 4 He believes that the number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) He randomly selects 100 litters each of size 4 and records the number of female cubs. The results are recorded in the table below.

He calculated the expected frequencies as follows

1. Find the value of \(r\) and the value of \(s\)
2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the number of female cubs in a litter can be modelled by \(\mathrm { B } ( 4,0.5 )\) You should clearly state your hypotheses and the critical value used.

13 In this question you must show detailed reasoning. The probability that Paul's train to work is late on any day is 0.15 , independently of other days.

1. The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable \(Y\). Find a value of \(a\) such that \(\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }\). In the expansion of \(( 0.15 + 0.85 ) ^ { 50 }\), the terms involving \(0.15 ^ { r }\) and \(0.15 ^ { r + 1 }\) are denoted by \(T _ { r }\) and \(T _ { r + 1 }\) respectively.
2. Show that \(\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }\).
3. The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable \(X\).
   1. Find the values of \(r\) for which \(\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )\).
   2. Hence find the most likely number of days on which the train will be late during a 50-day period.

9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.

1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.

14 A counter is initially at point \(O\) on the \(x\)-axis. A fair coin is thrown 6 times. Each time the coin shows heads, the counter is moved one unit in the positive \(x\)-direction. Each time the coin shows tails, the counter is moved one unit in the negative \(x\)-direction. The final distance of the counter from \(O\), in either direction, is denoted by \(D\). Determine the most probable value of \(D\). \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

10

1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
   Give your answer in ascending powers of \(x\).
2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
   The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.

3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.

1. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
   The random variable \(X\) is the number of yellow marbles selected.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.

1. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
2. Find the probability that the first wet day in October is 8 October.
3. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.