Question

A city has just added 110 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions: (i) Each new recruit has a 0.3 probability of remaining with the police force until retirement. (ii) Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.35. (iii) The number of pensions that the city will provide on behalf of each new hire is independent of the number of pensions it will provide on behalf of any other new hire. Determine the probability that the city will provide at most 62 pensions to the 110 new hires and their husbands. Enter your answer as a number accurate to 4 decimal places.

Answer 1

A city has just added 110 new female recruits to its police force.

there are 3 types of 110 recruits contributes either 0, 1, or 2 pensions.

1)The case of 0 pensions occurs when the employee leaves the force before retirement

has probability 1 − 0.3 = 0.7 ( Since we have given that Each new recruit has a 0.3 probability of remaining with the police force until retirement)

2)The case of 1 pension occurs if the recruit stays in the force until retirement, but remains unmarried;

this case has probability 0.3*0.35 = 0.105 (Since Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.35. )

3)The remaining case of 2 pensions occurs when the recruit stays in the force until retirement and is married at retirement; this case has probability 1 − 0.7 − 0.105 = 0.195

We have to find  the probability that the city will provide at most 62 pensions to the 110 new hires and their husbands.

Let X1, . . . , X110 denote the numbers of pensions contributed by the 110 recruits, then each Xi is a random variable with the above distribution, and mean and variance given by

µ = E(Xi) = 0 · 0.7 + 1 · 0.105 + 2 · 0.195 = 0.495,

σ ² = 0² · 0.7 + 1² · 0.105 + 2² · 0.195 − µ ² = 0.885 - 0.2450

= 0.64

The total number, say X, of pensions that the city has to provide then is the sum of these Xi ’s.

By the Central Limit Therom,

X is approximately normal with mean 110 * 0.495 = 45.45 and standard deviation √ 110 *√ 0.64 = 88. Thus, P(X ≤ 62) ≈

= Φ(0.08579)