Final answer:
For the given exponential distribution with a mean of 15 years, the value of λ is found to be 1/15. The probability that a randomly selected person still has vaccine-induced immunity 10 years after receiving their shot is e^(-2/3). The probability that both individuals have vaccine-induced immunity five years after receiving their booster is e^(-2/3).
Step-by-step explanation:
(a) To find the value of λ for the exponential distribution, we need to use the mean of 15 years. The mean of an exponential distribution is equal to 1/λ. So, 1/λ = 15, which means λ = 1/15.
(b) The probability that a randomly selected person still has vaccine-induced immunity 10 years after receiving their shot can be found using the exponential distribution. The probability can be calculated using the formula P(X > t) = e^(-λt), where X is the length of time of vaccine-induced immunity and t is the given time. Plugging in the values, we get P(X > 10) = e^(-1/15 * 10) = e^(-2/3).
(c) To find the probability that both individuals have vaccine-induced immunity five years after receiving their booster, we can use the same formula as in part (b) for each individual. The probability for one individual is P(X > 5) = e^(-1/15 * 5) = e^(-1/3). Since the events are independent, the probability that both individuals have vaccine-induced immunity is the product of their individual probabilities, which is e^(-1/3) * e^(-1/3) = e^(-2/3).