5.02k Calculate Poisson probabilities

Naomi produces oak tabletops, each of area 4·8 m². Defects in the oak tabletops occur randomly at a rate of 0·25 per m².

1. Find the probability that a randomly chosen tabletop will contain at most 2 defects. [3]
2. Find the probability that, in a random sample of 7 tabletops, exactly 4 will contain at most 2 defects each. [3]

A baker sells 3-5 birthday cakes per hour on average.

1. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution. [1]
2. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period. [2]
3. Calculate the probability that, during a randomly selected 3-hour period, the baker sells more than 10 birthday cakes. [3]
4. The baker sells a birthday cake at 9:30 a.m. Calculate the probability that the baker will sell the next birthday cake before 10:00 a.m. [3]
5. Select one of the assumptions in part (a) and comment on its reasonableness. [1]

Cars arrive at random at a toll bridge at a mean rate of 15 per hour.

1. Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval. [1]
2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive. [3]
3. Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3. [3]

Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).

1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
2. A customer arrives at exactly 11 a.m.
   1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
   2. Hence find the probability density function, \(f(t)\), of \(T\).
   3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]

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

1. Find the probability that at least 28 orders are received in a randomly chosen 10-minute period. [2]
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]
3. State a necessary assumption for the validity of your calculation in part (b). [1]

A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.

1. Find the probability of exactly 4 faults in a 15 metre length of cloth. (2)
2. Find the probability of more than 10 faults in 60 metres of cloth. (3)

A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80

1. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. (4)

The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.

1. Find the retailer's expected profit. (4)

Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.

1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. [3]

Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.

1. Find the probability that in exactly 3 of these periods there were no calls. [2]

On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.

1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]

The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520. [3]
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval. [1]

Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

1. State these two assumptions. [2]
2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).

1. 1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
   2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]

A certain type of fossil occurs at a mean rate of \(0.5\) per square metre at a particular location.

1. State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
2. Find the probability of at least 7 of these fossils occurring in an area of \(10 \text{ m}^2\). [2]
3. Given that at least 4 such fossils have occurred in an area of \(5 \text{ m}^2\), find the probability that there will be more than 6 found in this area of \(5 \text{ m}^2\). [3]
4. Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than \(0.999\). Give your answer to the nearest square metre. [4]