The mean number of automobiles entering a mountain tunnel per two-minute period is one. an excessive number of cars entering the tunnel during a brief period of time produces a hazardous exercises 137 situation. find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. does the poisson model seem reasonable for this problem?

A Poisson model seems reasonable for this problem, since we have the mean during the time interval.

There is a 1.9% probability that the number of autos entering the tunnel during a two-minute period exceeds three.

Step-by-step explanation:

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

In which

x is the number of sucesses

e = 2.71828 is the Euler number

is the mean in the given time interval.

The mean number of automobiles entering a mountain tunnel per two-minute period is one.

This means that .

For a Poisson model to be reasonable, we only need the mean during the time interval. So yes, a Poisson model seems reasonable for this problem.

Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three.

We want to find

Either this number is less or equal to 3, or it exceeds 3. The sum of the probabilities is decimal 1. So:

In which

So

Finally

There is a 1.9% probability that the number of autos entering the tunnel during a two-minute period exceeds three.