The number of times a hotline receives calls per minute obeys a Poisson distribution with a parameter of 4. Find the probability that exactly 8 calls are received in a certain minute.(The result is accurate to 4 decimal places)
Question
The number of times a hotline receives calls per minute obeys a Poisson distribution with a parameter of 4. Find the probability that exactly 8 calls are received in a certain minute.(The result is accurate to 4 decimal places)
Solution
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 = 4 calls per minute
- k = 8 (we want the probability of receiving exactly 8 calls in a certain minute)
Substituting these values into the formula, we get:
P(X=8) = (4^8 * e^-4) / 8!
Calculating the above expression gives a probability of approximately 0.0573 when rounded to four decimal places.
So, the probability that exactly 8 calls are received in a certain minute is 0.0573.
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