Question

The number of cars arriving at a toll booth in five-minute intervals is poisson distributed with a mean of 3 cars arriving in five-minute time intervals. the probability of 5 cars arriving over a five-minute interval is _______.​

Answer 1

`P(k=5) = (\lambda^k e^(-\lambda))/(k!) = (3^5 e^(-3))/(5!) = 0.1008`

Answer 2

The probability of 5 cars arriving over a five-minute interval is 0.1008 = 10.08%

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 time interval.

Mean of 3 cars arriving in five-minute time intervals.

This means that
`\mu = 3`

The probability of 5 cars arriving over a five-minute interval is

This is P(X = 5).

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

`P(X = 5) = (e^(-3)*3^(5))/((5)!) = 0.1008`

So

The probability of 5 cars arriving over a five-minute interval is 0.1008 = 10.08%