Final answer:
To find the probability of exactly two claims for a policy with at least one claim and a mean of 0.4 in a Poisson distribution, we use the Poisson probability mass function and adjust for the given condition.
Step-by-step explanation:
The given problem deals with a Poisson distribution, where the mean number of claims per policy in a year is 0.4. When considering a policy that has at least one claim, we are interested in finding the probability that there are exactly two claims. This situation is a conditional probability problem and requires the use of the Poisson probability mass function (pmf).
To find the probability of exactly two claims given at least one claim, we first find the unconditional probability of exactly two claims using the Poisson pmf, which is given by the formula P(X = k) = (e-λλk)/k!, where λ is the mean number of claims (0.4 in this case) and k is the exact number of claims we're finding the probability for. However, since we already know that there is at least one claim, we must adjust this probability to account for this condition, leading to a formal application of conditional probability formulas.
Knowing how to apply the Poisson distribution is crucial for various real-world situations, such as calculating insurance premiums, understanding event frequencies in a fixed interval, and approximating the binomial distribution in certain conditions. These examples showcase the practical applications of the Poisson distribution in fields such as insurance and risk management.