Basic sum of two Poissons

6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that

1. exactly 3 people arrive at the checkout during a 5 -minute period,
2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.

5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.

1. Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
2. Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.
   In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
3. Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.

3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.

1. Find the probability that more than 3 major avalanches occur in a 3-month period.
2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.

1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .

4 Bacteria of a certain type are randomly distributed in the water in two ponds, \(A\) and \(B\). The average numbers of bacteria per \(\mathrm { cm } ^ { 3 }\) in \(A\) and \(B\) are 0.32 and 0.45 respectively.

1. Samples of \(8 \mathrm {~cm} ^ { 3 }\) of water from \(A\) and \(12 \mathrm {~cm} ^ { 3 }\) of water from \(B\) are taken at random. Find the probability that the total number of bacteria in these samples is at least 3 .
2. Find the probability that in a random sample of \(155 \mathrm {~cm} ^ { 3 }\) of water from \(A\), the number of bacteria is less than 35 .

1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .

1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .

6 In their football matches, Rovers score goals independently and at random times. Their average rate of scoring is 2.3 goals per match.

1. State the expected number of goals that Rovers will score in the first half of a match.
2. Find the probability that Rovers will not score any goals in the first half of a match but will score one or more goals in the second half of the match.
3. Football matches last for 90 minutes. In a particular match, Rovers score one goal in the first 30 minutes. Find the probability that they will score at least one further goal in the remaining 60 minutes. Independently of the number of goals scored by Rovers, the number of goals scored per football match by United has a Poisson distribution with mean 1.8.
4. Find the probability that a total of at least 3 goals will be scored in a particular match when Rovers play United.

1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .

1 A car repair firm receives call-outs both as a result of breakdowns and also as a result of accidents. On weekdays (Monday to Friday), call-outs resulting from breakdowns occur at random, at an average rate of 6 per 5 -day week; call-outs resulting from accidents occur at random, at an average rate of 2 per 5 -day week. The two types of call-out occur independently of each other. Find the probability that the total number of call-outs received by the firm on one randomly chosen weekday is more than 3 .

1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.

1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
2. State one assumption needed in the calculation.

1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.

2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

4 The number of printers, \(V\), bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, \(W\), bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4

1. Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
   [0pt] [2 marks] 4
2. State the circumstance under which the distributional model you used in part (a) would not be valid.
   [0pt] [1 mark]

2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

2 A new information technology centre is advertising places on its one-week residential computer courses.

1. The number of places, \(X\), booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
   1. State the standard deviation of \(X\).
   2. Calculate \(\mathrm { P } ( 6 < X < 12 )\).
2. The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
   1. Write down the distribution for \(T\), the total number of places booked each week on the publishing and web design courses.
   2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
3. The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution \(\mathrm { Po } ( 12.0 )\) could provide a suitable model for the number of places booked each week on the database course.

1 The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

1. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
2. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
   1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
   2. What assumption did you make in answering part (b)(i)?

5

1. The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
   1. at least 9 minor accidents;
   2. more than 5 but fewer than 10 minor accidents.
2. The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
3. The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
   1. state the value of \(\lambda\);
   2. determine \(\mathrm { P } ( T > 16 )\);
   3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.

1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
2. A quarter of a game lasts 12 minutes.
   1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
   2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

6 A company owns two machines, \(A\) and \(B\), which make toys. Both machines run continuously and independently. Machine \(A\) makes an average of 2 errors per hour.
6

1. Using a Poisson model, find the probability that the machine makes exactly 5 errors in 4 hours, giving your answer to three significant figures. 6
2. Machine \(B\) makes an average of 5 errors per hour. Both machines are switched on and run for 1 hour. The company finds the probability that the total number of errors made by machines \(A\) and \(B\) in 1 hour is greater than 8 . If the probability is greater than 0.4 , a new machine will be purchased.
   Using a Poisson model, determine whether or not the toy company will purchase a new machine.
   6
3. After investigation, the standard deviation of errors made by machine \(A\) is found to be 0.5 errors per hour and the standard deviation of errors made by machine \(B\) is also found to be 0.5 errors per hour. Explain whether or not the use of Poisson models in parts (a) and (b) is appropriate.

1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\mathrm { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).