5. Binomial Distribution & Discrete Random Variables
Poisson Distribution
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A small electronics retailer tracks the number of customers who arrive to purchase replacement phone chargers. Based on historical data, the store finds that, on average, 3 customers per day buy a charger. The store manager wants to use this information to optimize inventory decisions and reduce the risk of stockouts.
(A) Find the probability that 5 customers buy a charger in a given day.
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Verified step by step guidance1
Identify the type of probability distribution: Since the problem involves counting the number of events (customers buying chargers) in a fixed interval (1 day) with a known average rate (3 customers per day), this is a Poisson distribution problem.
Write the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate (mean), k is the number of occurrences, and e is the base of the natural logarithm (approximately 2.718).
Substitute the given values into the formula: Here, λ = 3 (average number of customers per day) and k = 5 (the number of customers we are finding the probability for). The formula becomes P(X = 5) = (3^5 * e^(-3)) / 5!.
Simplify the components of the formula: Calculate 3^5 (3 raised to the power of 5), e^(-3) (exponential of -3), and 5! (factorial of 5). These values will be used to compute the probability.
Combine the results: Multiply the numerator (3^5 * e^(-3)) and divide by the denominator (5!) to find the probability P(X = 5). This will give the final probability value.
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