Single batch expected count

5 Douglas plays darts, and the probability that he hits the number he is aiming at is 0.87 for any particular dart. Douglas aims a set of three darts at the number 20; the number of times he is successful can be modelled by \(\mathrm { B } ( 3,0.87 )\).

1. Calculate the probability that Douglas hits 20 twice.
2. Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
3. Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.

4 A small business has 8 workers. On a given day, the probability that any particular worker is off sick is 0.05 , independently of the other workers.

1. A day is selected at random. Find the probability that
   (A) no workers are off sick,
   (B) more than one worker is off sick.
2. There are 250 working days in a year. Find the expected number of days in the year on which more than one worker is off sick.

1 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.

1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.

3 Douglas plays darts, and the probability that he hits the number he is aiming at is 0.87 for any particular dart. Douglas aims a set of three darts at the number 20 ; the number of times he is successful can be modelled by \(\mathrm { B } ( 3,0.87 )\).

1. Calculate the probability that Douglas hits 20 twice.
2. Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
3. Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.

4 At a call centre, \(85 \%\) of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

1. Find the probability that exactly 29 of these callers are put on hold.
2. Find the probability that at least 29 of these callers are put on hold.
3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

5 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.

1. Find the probability that in a batch of 50 there is
   (A) exactly one faulty teapot,
   (B) more than one faulty teapot.
2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.

2 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

8 At a call centre, 85\% of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

1. Find the probability that exactly 29 of these callers are put on hold.
2. Find the probability that at least 29 of these callers are put on hold.
3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

2 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.

1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.

4 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.

1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.

3 At a call centre, \(85 \%\) of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

1. Find the probability that exactly 29 of these callers are put on hold.
2. Find the probability that at least 29 of these callers are put on hold.
3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.

1. Find the probability that in a batch of 50 there is
   (A) exactly one faulty teapot,
   (B) more than one faulty teapot.
2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.

15 Ali and Sam are playing a game in which Ali tosses a coin 5 times. If there are 4 or 5 heads, Ali wins the game. Otherwise Sam wins the game. They decide to play the game 50 times.

1. Initially Sam models the situation by assuming the coin is fair. Determine the number of games Ali is expected to win according to this model. Ali thinks the coin may be biased, with probability \(p\) of obtaining heads when the coin is tossed. Before playing the game, Ali and Sam decide to collect some data to estimate the value of \(p\). Sam tosses the coin 15 times and records the number of heads obtained. Ali tosses the coin 25 times and records the number of heads obtained.
2. Explain why it is better to use the combined data rather than just Sam's data or just Ali's data to estimate the value of \(p\). Ali records 20 heads and Sam records 8 heads.
3. Use the combined data to estimate the value of \(p\). Ali now models the situation using the value of \(p\) found in part (c) as the probability of obtaining heads when the coin is tossed.
4. Determine how many games Ali expects to win using this value of \(p\) to model the situation.
5. Ali wins 25 of the 50 games. Explain whether Sam's model or Ali's model is a better fit for the data. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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