Answer:
To compute the probabilities for each scenario, we will use the Poisson distribution formula. The formula for the probability mass function of a Poisson distribution is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
P(X = k) is the probability of getting exactly k hits
λ is the average rate of hits per unit of time
k is the number of hits we are interested in
For scenario (a) - exactly five hits:
We need to calculate P(X = 5) for a time interval of 9 minutes (7:34 P.M. to 7:43 P.M.) with an average rate of 0.3 hits per minute.
λ = 0.3 * 9 = 2.7
P(X = 5) = (e^(-2.7) * 2.7^5) / 5!
For scenario (b) - fewer than five hits:
We need to calculate the cumulative probability from k = 0 to k = 4.
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
For each k value, we can use the formula P(X = k) as mentioned above with λ = 2.7.
For scenario (c) - at least five hits:
We need to calculate the cumulative probability from k = 5 to k = infinity.
P(X ≥ 5) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))
After calculating the probabilities for each scenario, we can interpret the results. For example, if the probability of exactly five hits is high, it means that it is likely to have exactly five hits during the given time interval. If the probability of fewer than five hits is high, it means that it is likely to have fewer than five hits during the given time interval. And if the probability of at least five hits is high, it means that it is likely to have five or more hits during the given time interval.
I hope this helps :)