Final answer:
To find the probability that the lifetime of at least one component exceeds 3, we can use the exponential distribution formula. The probability can be calculated by subtracting the probability that all components last less than or equal to 3 from 1. The rate parameter of the exponential distribution is 1/10, since the average lifetime is 10 years.
Step-by-step explanation:
To find the probability that the lifetime of at least one component exceeds 3, we need to find the complementary probability that the lifetime of all components is less than or equal to 3 and then subtract it from 1. Since the lifetime of each component is exponentially distributed with an average of 10 years, we can use the exponential distribution formula to calculate the probability of a component lasting less than or equal to 3 years.
The formula is: P(X <= x) = 1 - e^(-λx), where x is the value we're interested in and λ is the rate parameter of the exponential distribution. In this case, λ is equal to 1/10 since the average lifetime is 10 years. So, the probability that a component lasts less than or equal to 3 years is P(X <= 3) = 1 - e^(-1/10 * 3). To find the probability that at least one component lasts more than 3 years, we subtract this probability from 1:
P(X > 3) = 1 - P(X <= 3).