Final answer:
The probability that both Mr. and Mrs. Smith will be alive in 10 years is 42%, that Mr. Smith is not living while Mrs. Smith is living is 28%, and the probability that at least one of them is living in 10 years is 88%.
Step-by-step explanation:
The question regarding Mr. and Mrs. Smith is focused on determining various probabilities related to their life expectancy after retirement. To achieve this, we need to use the rules of probability. Specifically, we apply the multiplication rule of independent events and the complement rule:
- a) The probability that both Mr. and Mrs. Smith will be alive for 10 years from now is found by multiplying the individual probabilities: 0.60 (for Mr. Smith) × 0.70 (for Mrs. Smith) = 0.42, or 42%.
- b) The probability that in 10 years, Mr. Smith is not living and Mrs. Smith is living is found by multiplying the probability of Mr. Smith not living (1 - 0.60 = 0.40) with the probability of Mrs. Smith living (0.70): 0.40 × 0.70 = 0.28, or 28%.
- c) The probability that in 10 years at least one is living involves calculating the complement of the probability that both are not living. The probability that both are not living is (1 - 0.60) × (1 - 0.70) = 0.12 or 12%. Therefore, the probability that at least one is living is 1 - 0.12 = 0.88, or 88%.