A park receives an average of 20 tourists per hour. Assuming that the arrival time of tourists follows a Poisson distribution, what is the probability that exactly 8 tourists will be received in any given hour?Round your results to four decimal places.
Question
A park receives an average of 20 tourists per hour. Assuming that the arrival time of tourists follows a Poisson distribution, what is the probability that exactly 8 tourists will be received in any given hour?Round your results to four 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 = 20 tourists per hour
- k = 8 (we want the probability of receiving exactly 8 tourists in a given hour)
Substituting these values into the formula, we get:
P(X=8) = (20^8 * e^-20) / 8!
Calculating the above expression gives a probability of approximately 0.0258 when rounded to four decimal places.
So, the probability that exactly 8 tourists will be received in any given hour is 0.0258.
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