At a dress shop, it is noticed that 120 people are arriving in 2 hours. Given this what is the probability that, during 1 minute period, exactly 5 people arrive. Use Poisson Distribution and round your answer to four decimal places.
Question
At a dress shop, it is noticed that 120 people are arriving in 2 hours. Given this what is the probability that, during 1 minute period, exactly 5 people arrive. Use Poisson Distribution and round your answer to four decimal places.
Solution
To solve this problem, we need to use the Poisson distribution. The Poisson distribution is used to model the number of events happening in a fixed interval of time or space. The formula for the Poisson distribution is:
P(X=k) = (λ^k * e^-λ) / k!
Where:
- λ is the average rate of occurrence (in this case, the average number of people arriving per minute)
- k is the number of occurrences we are interested in (in this case, 5 people)
- e is the base of the natural logarithm (approximately 2.71828)
- "!" denotes a factorial, meaning the product of all positive integers up to that number.
First, we need to find λ. We know that 120 people arrive in 2 hours, which is 120 minutes. So, λ = 120 people / 120 minutes = 1 person per minute.
Substituting these into the formula gives:
P(X=5) = (1^5 * e^-1) / 5! = (1 * 0.3679) / 120 = 0.003065
Rounding to four decimal places, the probability that exactly 5 people arrive in a 1 minute period is approximately 0.0031.
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