Single period normal approximation - large lambda direct

6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of

1. exactly 4 calls in an 8 -minute period,
2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.

5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).

1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.

An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]

Find the probability that in a randomly chosen hour

1. all Internet users connect at their first attempt, [2]
2. more than 4 users fail to connect at their first attempt. [2]

1. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
2. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [6]

Minor defects occur in a particular make of carpet at a mean rate of 0.05 per m\(^2\).

1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer.

A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires 30 m\(^2\) of this carpet. Find the probability that the foyer carpet contains

1. exactly 2 defects, [3]
2. more than 5 defects. [3]

The carpet fitter orders a total of 355 m\(^2\) of the carpet for the whole hotel.

1. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects. [6]

An estate agent sells properties at a mean rate of 7 per week.

1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model. [3]
2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties. [2]
3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties. [6]

In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km\(^2\).

1. Find the probability that in a randomly selected area of 1000 km\(^2\) there are at least 10 barn owls. [2]
2. Find the probability that in a randomly selected area of 200 km\(^2\) there are exactly 2 barn owls. [3]
3. Using a suitable approximation, find the probability that in a randomly selected area of 50000 km\(^2\) there are at least 470 barn owls. [6]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]