N2013 P2 Q12

A company has two departments and each department records the number of employees absent through illness each day. Over a long period of time it is found that the average numbers of employees absent on a day are \(1.2\) for the Administration Department and \(2.7\) for the Manufacturing Department.

1. State, in this context, two conditions that must be met for the numbers of absences to be well modelled by Poisson distributions. Explain why each of your two conditions may not be met.[3]

For the remainder of this question assume that these conditions are met. You should assume also that absences in the two departments are independent of each other.

2. Find the smallest number of days for which the probability that no employee is absent through illness from the Administration Department is less than \(0.01\).[2]

Each employee absent on a day represents one ‘day of absence’. So, one employee absent for \(3\) days contributes \(3\) days of absence, and \(5\) employees absent on \(1\) day contribute \(5\) days of absence.

3. Find the probability that, in a \(5\)-day period, the total number of days of absence in the two departments is more than \(20\).[3]
4. Use a suitable approximation, which should be stated together with its parameter(s), to find the probability that, in a \(60\)-day period, the total number of days of absence in the two departments is between \(200\) and \(250\) inclusive.[4]