A city has just added 100 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:

1. Each new recruit has a 0.4 probability of remaining with the police force until retirement.
2. Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.25.
3. The events of different new hires reaching retirement and the events of different new hires being married at retirement are all mutually independent events.

Calculate the probability that the city will provide at most 90 pensions to the 100 new hires and their husbands.

• 0.60
• 0.67
• 0.75
• 0.93
• 0.99

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

For a single recruit, the probability of 0 pensions is 0.6, of 1 pension is 0.4(0.25) = 0.1, and of 2 pensions is 0.4(0.75) = 0.3. The expected number of pensions is 0(0.6) + 1(0.1) + 2(0.3) = 0.7. The second moment is 0(0.6) + 1(0.1) + 4(0.3) = 1.3. The variance is 1.3 – 0.49 = 0.81. For 100 recruits the mean is 70 and the variance is 81. The probability of providing at most 90 pensions is (with a continuity correction) the probability of being below 90.5. This is (90.5 – 70)/9 = 2.28 standard deviations above the mean. From the tables, this probability is 0.9887.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.