In a library, there will be 7 people asking questions to the librarian every hour. What is the probability that, during a 10-minute period, the librarian will have exactly one query (round your answer to four decimal places)?
Question
In a library, there will be 7 people asking questions to the librarian every hour. What is the probability that, during a 10-minute period, the librarian will have exactly one query (round your answer to four decimal places)?
Solution
This is a Poisson distribution problem. The Poisson distribution gives the probability of a given number of events (in this case, queries to the librarian) happening in a fixed interval of time (in this case, 10 minutes) when these events occur with a known average rate (in this case, 7 queries per hour) and independently of the time since the last event.
First, we need to find the average rate of queries per 10 minutes. If there are 7 queries per hour (60 minutes), then there are (7 queries / 60 minutes) * 10 minutes = 1.167 queries per 10 minutes.
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 events per interval
- e is the base of the natural logarithm, approximately equal to 2.71828
- k! is the factorial of k
In this case, we want to find the probability of exactly one query (k=1) during a 10-minute period, with an average rate of 1.167 queries per 10 minutes (λ=1.167).
So, we plug these values into the formula:
P(X=1) = (1.167^1 * e^-1.167) / 1! = 1.167 * e^-1.167 = 0.3144 (rounded to four decimal places)
So, the probability that, during a 10-minute period, the librarian will have exactly one query is approximately 0.3144.
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