Students arrive at Central Library at an average of 15 per quarter of the lunch hour. What is the probability that the next student will arrive within 5 minutes? ( round your answer to four decimal places)
Question
Students arrive at Central Library at an average of 15 per quarter of the lunch hour. What is the probability that the next student will arrive within 5 minutes? ( round your answer to four decimal places)
Solution
This is a Poisson distribution problem. The Poisson distribution is used to model the number of events occurring within a given time period.
The formula for the Poisson probability is:
P(X=k) = (λ^k * e^-λ) / k!
where:
- P(X=k) is the probability of k events in the interval
- λ is the average rate of value
- e is the base of the natural logarithm, approximately equal to 2.71828
- k is the actual number of successes
In this case:
- λ = 15 students per 15 minutes * (5 minutes / 15 minutes) = 5 students
- k = 1 (we want the probability of one student arriving in a 5-minute period)
Substituting these values into the formula, we get:
P(X=1) = (5^1 * e^-5) / 1!
Calculating the above expression gives a probability of approximately 0.0337 when rounded to four decimal places.
So, the probability that the next student will arrive within 5 minutes is 0.0337.
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