5. Binomial Distribution & Discrete Random Variables
Poisson Distribution
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A baker wants to predict how many customers will enter their bakery. On average, 2 customers come into the bakery every 15 minutes. Find the probability that exactly 5 customers will enter the bakery
(B) 4 or fewer customers enter the bakery in a random 15 min period.
A
B
C
D
Verified step by step guidance1
Step 1: Recognize that this is a Poisson distribution problem. The Poisson distribution is used to model the number of events (e.g., customers entering a bakery) occurring in a fixed interval of time or space, given a known average rate (λ). Here, the average rate (λ) is 2 customers per 15 minutes.
Step 2: Recall the Poisson probability formula: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (e.g., customers), λ is the average rate, and e is the mathematical constant approximately equal to 2.718.
Step 3: For part (A), calculate the probability of exactly 5 customers entering the bakery in 15 minutes. Substitute λ = 2 and k = 5 into the formula: P(X = 5) = (2^5 * e^(-2)) / 5!. Simplify the expression step by step to find the probability.
Step 4: For part (B), calculate the probability of 4 or fewer customers entering the bakery. This is the cumulative probability P(X ≤ 4), which is the sum of probabilities for X = 0, 1, 2, 3, and 4. Use the Poisson formula for each value of k (0 through 4) and sum the results: P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Step 5: After calculating the individual probabilities for part (B), add them together to find the cumulative probability. Compare the results to the provided answer choices to identify the correct one.
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