Question

(10 points) It is known that cars arrive at a drive-through at a rate of three cars per minute between 12 noon and 1:00 pm. Assuming the number of cars that arrive in any time interval follows a Poisson distribution, what is the probability that exactly 9 cars arrive between 12:10 and 12:20. (Note: the question gives the rate is

Answer 1

0.000005 probability that exactly 9 cars arrive between 12:10 and 12:20.

Explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\mu` is the mean in the given interval.

Mean of 3 cars per minute:

So, between 12:10 and 12:20, there is an interval of 10 minutes, which means that
`\mu = 3*10 = 30`

What is the probability that exactly 9 cars arrive between 12:10 and 12:20?

This is
`P(X = 9)`. So

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

`P(X = 9) = (e^(-30)*30^(9))/((9)!) = 0.000005`

0.000005 probability that exactly 9 cars arrive between 12:10 and 12:20.