Question

The number of hits on a certain website during a one-minute interval follows a Poisson distribution with a mean rate of four hits per minute. What is the probability that there is at least one hit in a 30-second period (that is the probability of one or more hits)

Answer 1

86.47% probability that there is at least one hit in a 30-second period

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 rate of four hits per minute.

This means that
`\mu = 4n`, in which n is the number of minutes.

What is the probability that there is at least one hit in a 30-second period

30 seconds is 0.5 minutes, so
`\mu = 4*0.5 = 2`

Either the site doesn't get a hit during this period, or it does. The sum of the probabilities of these events is 1. So

`P(X = 0) + P(X \geq 1) = 1`

We want
`P(X \geq 1)`

Then

`P(X \geq 1) = 1 - P(X = 0)`

In which

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

`P(X = 0) = (e^(-2)*2^(0))/((0)!) = 0.1353`

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.1353 = 0.8647`

86.47% probability that there is at least one hit in a 30-second period