5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

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 \(\mathrm{Po}(0.8)\).

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 \(\mathrm{Po}(1.5)\). 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]

The numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\)? [2]

Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).

1. Find
   1. P\((X = 6)\), [2]
   2. P\((X \geqslant 8)\). [2]

The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).

1. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
2. State an assumption needed for your answer to part (c) to be valid. [1]
3. Give a reason why the assumption in part (d) may not be valid in practice. [1]

1. State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day. [2]

Jane records the number of junk emails which she receives each day. During working hours (\(9\)am to \(5\)pm, Monday to Friday) the mean number of junk emails is \(7.4\) per day. Outside working hours (\(5\)pm to \(9\)am), the mean number of junk emails is \(0.3\) per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".

1. Find the probability that the number of junk emails which she receives between \(9\)am and \(5\)pm on a Monday is
   1. exactly \(10\), [1]
   2. at least \(10\). [2]
2. 1. What assumption must you make to calculate the probability that the number of junk emails which she receives from \(9\)am Monday to \(9\)am Tuesday is at most \(20\)? [1]
   2. Find the probability. [2]

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)

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]

The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. [2]

Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).

1. 1. Write down an expression for P\((X = r)\). [1]
   2. Hence find P\((X = 3)\). [1]
2. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match. [4]

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]