Multi-period repeated application

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.

1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
2. there will be 2 accidents in the same week,
3. there is at least one accident per week for 3 consecutive weeks,
4. there are more than 4 accidents in a 2 week period.

2. On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407 , to 3 significant figures. Find the probability of
2. exactly 2 accidents in the next 6 month period,
3. no accidents in exactly 2 of the next 4 months.

1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .

1. 1. Calculate \(\mathrm { P } ( X = 2 )\).
   2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
2. 1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
   2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

2 The number of computers, \(A\), bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, \(B\), bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .

1. 1. Calculate \(\mathrm { P } ( A = 4 )\).
   2. Determine \(\mathrm { P } ( B \leqslant 6 )\).
   3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
2. Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
3. The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
   1. Calculate the mean and variance of these data.
   2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.

1 The number of cars, \(X\), passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.

1. Calculate \(\mathrm { P } ( X = 2 )\).
2. 1. Write down the distribution of \(Y\), the number of cars passing along this road in a 5-minute interval.
   2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.

2

1. The number of telephone calls, \(X\), received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine \(\mathrm { P } ( X = 8 )\).
2. The number of telephone calls, \(Y\), received per hour for Dr Bracken may be modelled by a Poisson distribution with mean \(\lambda\) and standard deviation 3 .
   1. Write down the value of \(\lambda\).
   2. Determine \(\mathrm { P } ( Y > \lambda )\).
3. 1. Assuming that \(X\) and \(Y\) are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
   2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
   3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

8 The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes. 8

1. Find the probability that:
   8
   1. (i) the office receives exactly 2 calls in 10 minutes; 8
   2. (ii) the office receives more than 30 calls in an hour.
      8
   3. The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods. Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.
      8
   4. The office has just received a call.
      8
   5. 1. Find the probability that the next call is received more than 10 minutes later.
         8
   6. (ii) Mahah arrives at the office 5 minutes after the last call was received.
      State the probability that the next call received by the office is received more than 10 minutes later. Explain your answer. \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-14_2492_1721_217_150} Additional page, if required.
      Write the question numbers in the left-hand margin. Additional page, if required.
      Write the question numbers in the left-hand margin.

4 In a Football League match, the number of goals scored by the home team can be modelled by the distribution \(\mathrm { Po } ( 2.4 )\). The number of goals scored by the away team can be modelled by the distribution Po(1.8).

1. State a necessary assumption for the total number of goals scored in one match to be modelled by the distribution \(\operatorname { Po } ( 4.2 )\).
2. Assume now that this assumption holds.
   1. Write down an expression for the probability that the total number of goals scored in \(n\) randomly chosen games is less than 4 .
   2. Find the probability that the result of a randomly chosen game is either 0-0 or 1-1.