Final answer:
The probability of exactly 5 cars showing up at the retailer between 2pm and 3pm next Monday, given that the average number of vehicles is 6.5 and follows a Poisson distribution, is calculated using the Poisson formula.
Step-by-step explanation:
The question pertains to the use of the Poisson distribution to determine the probability of a specific event. Since the average number of vehicles visiting the retailer between 2pm and 3pm each afternoon from Monday through Thursday is 6.5, we use this as the Poisson rate (λ) to calculate the probability that exactly 5 cars will show up next Monday.
To calculate this probability, the formula for the Poisson probability is:
P(X=k) = (e-λ * λk) / k!
Where:
- e is the base of the natural logarithm, approximately equal to 2.71828,
- λ (lambda) is the average number of events (in this case, λ = 6.5),
- k is the number of events we want to find the probability for (in this question, k = 5),
- k! is the factorial of k.
Applying these values to the formula, we get:
P(X=5) = (e-6.5 * 6.55) / 5!
By calculating this expression, we can find the exact probability of exactly 5 cars showing up.