Final answer:
To determine the probability of fewer than four claims being filed in the next week at a life insurance company, we use the Poisson distribution with the average rate of λ = 7 claims per week. The required probability is the sum of the probabilities of having 0, 1, 2, or 3 claims, each of which is calculated using the Poisson probability mass function.
Step-by-step explanation:
The question revolves around a scenario where a life insurance company experiences an average of seven claims per week at its Nashville branch. The student is interested in finding the probability that fewer than four claims will be filed in the following week. This is a classic example of a Poisson distribution, as it deals with the number of events (claims in this case) that occur within a fixed interval of time.
To calculate this probability, we sum the probabilities of having 0, 1, 2, or 3 claims in a week, since these are all the possible outcomes that are fewer than four.
Probability Calculation
Let X represent the number of claims filed in a week. Since 7 claims is the average, we use λ = 7. The probability mass function of a Poisson distribution is given by:
P(X = k) = λ^k * e^{-λ} / k!, where k is the number of occurrences.
To find the probability of at most three claims, we calculate:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
In practice, these calculations often require the use of statistical software or tables due to the factorial calculations involved.
Understanding the Poisson distribution and its applications can be essential for various professions, including actuaries and insurance analysts, who apply mathematical and statistical methods to assess risk in insurance policies and finance.