The probability that a 50 year old man will be alive in 10 years is 0.78 , and the probability that his 45 year old wife will be alive is 0.91 . Determine the probability that in 10 years:
a. both will be alive
b. at least one will be alive (HINT: what is the complement to the event that at least one is alive?)
c. only the wife will be alive
a. 0.78 * 0.91
b. both dead = (1 - .78)(1 - .91)
so at least one alive = 1 - (1 - .78)(1 - .91)
c. her alive * him dead = .91 (1 - .78)
.... and it looks like my wife and I are long gone.
To determine the probabilities of these events, we can use the concept of independent events and the multiplication rule.
a. To calculate the probability that both the man and his wife will be alive in 10 years, we multiply their individual probabilities of being alive: 0.78 * 0.91 = 0.7098. Therefore, the probability that both will be alive is 0.7098 or approximately 0.71.
b. To calculate the probability that at least one of them will be alive, we can calculate the complement of the event that neither of them will be alive. The complement of an event (A) is the probability of the opposite event, denoted by A'. Thus, the probability of at least one of them being alive is equal to 1 - (the probability that neither of them is alive).
The probability that neither the man nor his wife will be alive in 10 years is found by multiplying the probabilities of each of them not being alive: (1 - 0.78) * (1 - 0.91) = 0.22 * 0.09 = 0.0198.
Therefore, the probability that at least one of them will be alive is 1 - 0.0198 = 0.9802 or approximately 0.98.
c. To calculate the probability that only the wife will be alive in 10 years, we need to find the probability of the man not being alive and the wife being alive. This is given by: (1 - 0.78) * 0.91 = 0.22 * 0.91 = 0.2002.
Therefore, the probability that only the wife will be alive is approximately 0.20 or 20%.