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
(A) Exactly 4 customers will enter the bakery between 9:00 – 9:15.
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Verified step by step guidance1
Step 1: Recognize that this problem involves a Poisson distribution. The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence (λ).
Step 2: Determine the average rate (λ) for the interval. The problem states that 2 customers enter the bakery every 15 minutes. For the interval between 9:00 and 9:15, λ = 2.
Step 3: Use the Poisson probability formula: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (customers in this case), λ is the average rate, and e is the mathematical constant approximately equal to 2.718.
Step 4: Substitute the values into the formula for the first part of the problem (exactly 5 customers): λ = 2, k = 5. Calculate the numerator (λ^k * e^(-λ)) and denominator (k!) separately.
Step 5: Repeat the process for the second part of the problem (exactly 4 customers between 9:00 and 9:15). Substitute λ = 2 and k = 4 into the Poisson formula, and calculate the probability using the same steps as above.
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