Sum of independent Poisson processes

5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.

1. Find the probability that more than 4 cars arrive during a 20-minute period.
2. Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.
   Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
3. Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .

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.

6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.

1. Find the probability that exactly 2 televisions arrive in a 2-day period.
2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
   Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.

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.

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 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:

1. a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
2. a 20 m length of tideline along this beach contains exactly 2 drinks cans;
3. a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
   [0pt] [4 marks]

1 The water in a pond contains three different species of a spherical green algae:
Volvox globator, at an average rate of 4.5 spheres per \(1 \mathrm {~cm} ^ { 3 }\);
Volvox aureus, at an average rate of 2.3 spheres per \(1 \mathrm {~cm} ^ { 3 }\);
Volvox tertius, at an average rate of 1.2 spheres per \(1 \mathrm {~cm} ^ { 3 }\).
Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond.
Find the probability that:

1. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains no more than 5 Volvox globator spheres;
2. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains at least 2 Volvox aureus spheres;
3. a \(5 \mathrm {~cm} ^ { 3 }\) sample contains more than 8 but fewer than 12 Volvox tertius spheres;
4. a \(0.1 \mathrm {~cm} ^ { 3 }\) sample contains a total of exactly 2 Volvox spheres;
5. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains at least 1 sphere of each of the three different species of algae.
   [0pt] [3 marks]

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. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8

The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3

1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
   State your hypotheses clearly.