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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Assurance and annuity functions involving two lives

Independent lives (x) and (y) have constant forces of mortality of 0.04 and 0.06 per year respectively. What is the probability that (x) dies before (y) and that this first death occurs within 10 years?

The probability is 0.2528. The first death happens within 10 years with probability 1 − e^{-1} = 0.6321, and it is (x)'s death with probability 0.04/0.10 = 0.4. Multiplying gives 0.4 × 0.6321 = 0.2528.

  1. A0.3793
  2. B0.4000
  3. C0.2528Correct
  4. D0.1319
  5. 0.6321

Explanation

The first death occurs at rate 0.10, so P(T_xy < 10) = 1 − e^{-1} = 0.6321. The probability that the first death is (x)'s is 0.04/0.10 = 0.4, independent of the time of first death. The product is 0.4 × 0.6321 = 0.2528. Using 0.6 gives 0.3793, which is the probability that (y) dies first within 10 years.

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