Final Answers:
a. P(X = 2) ≈ 0.2707
b. P(X = 4) ≈ 0.1954
c. P(X = 5) ≈ 0.1755
d. P(X = 1) ≈ 0.1839
Step-by-step explanation:
The Poisson distribution calculates the probability of a specific number of events occurring in a fixed interval of time or space, given the average rate of occurrence and assuming events happen independently. The formula to compute the probability for a Poisson distribution is P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of occurrence.
For these scenarios, each mean number of people audited serves as λ. Using the Poisson probability formula, we can calculate the probabilities:
a. For a mean of 2 people audited (λ = 2), the probability of exactly 2 people being audited is around 0.2707.
b. With a mean of 4 people audited (λ = 4), the probability of exactly 4 individuals being audited is approximately 0.1954.
c. At a mean of 5 individuals audited (λ = 5), the probability of exactly 5 people being audited is roughly 0.1755.
d. Finally, for a mean of 1 person audited (λ = 1), the probability of exactly 1 person being audited is about 0.1839.
These probabilities demonstrate the likelihood of specific numbers of individuals being audited among the randomly selected 100 people with tax returns over $25,000 in one year.