Direct binomial probability calculation

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
   1. exactly 4 red loom bands;
   2. at most 10 yellow loom bands;
   3. at least 30 blue or green loom bands;
   4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
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3 A tennis player is practising her serve. Each time she serves, she has a \(55 \%\) chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.

1. Find the probability that the player is not successful in any of her first three serves.
2. Determine the probability that the player is successful at least 10 times in her first 20 serves.
3. Determine the probability that the player is successful for the first time on her fifth serve.
4. Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability \(p\) of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
5. Determine the value of \(p\).

3. In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
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3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages. [1]
4. Determine the expected number of messages required for 3 of them to be misunderstood.
5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.
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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.

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.

7. An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

1. Find the probability that
   (A) exactly 8 of these orders are delivered within 24 hours,
   (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18 , find the critical region for this test, showing all of your calculations.
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