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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 survival probabilities p_x = 0.98, p_{x+1} = 0.97 and p_y = 0.96, p_{y+1} = 0.94. What is the probability that exactly one of the two lives is alive at the end of two years?

The probability is 0.1374. Two-year survival probabilities are 0.9506 for (x) and 0.9024 for (y). Exactly one alive is 0.9506 × 0.0976 + 0.0494 × 0.9024, which sums to about 0.1374. Both alive would be 0.8578.

  1. A0.8578
  2. B0.9952
  3. C0.1374Correct
  4. D0.0048
  5. 0.1000

Explanation

2p_x = 0.98 × 0.97 = 0.9506 and 2p_y = 0.96 × 0.94 = 0.9024. Exactly one alive = 0.9506 × 0.0976 + 0.0494 × 0.9024 = 0.09278 + 0.04458 = 0.1374. The value 0.8578 is the probability that both are alive.

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