Question

Toll booths on the New York State thruway are often congested because of the large number of cars waiting to pay. A consultant working for the state concluded that if service times are measured from the time a car stops in line until it leaves, service times are exponentially distributed with a mean of 2.7 minutes. What proportion of cars can get through the toll booth in less than 3 minutes?

Answer 1

`P(X < 3) = 67.14\%`

Therefore, 67.14% of the cars can get through the toll booth in less than 3 minutes.

Step-by-step explanation:

A consultant working for the state concluded that if service times are measured from the time a car stops in line until it leaves, service times are exponentially distributed with a mean of 2.7 minutes.

Let X be a random variable. The service time has an exponential distribution with a mean of 2.7 minutes.

Mean = μ = 2.7 minutes

Decay rate = m = 1 /μ

Decay rate = m = 1/2.7

Decay rate = m = 0.371

The cumulative probability distribution function is given by

`P(X < x) = 1 - e^(-mx)`

Where m is the decay rate and x < 3

So the proportion of cars that can get through the toll booth in less than 3 minutes is

`P(X < 3) = 1 - e^(-0.371(3))`

`P(X < 3) = 1 - 0.3286`

`P(X < 3) = 0.6714`

`P(X < 3) = 67.14\%`

Therefore, 67.14% of the cars can get through the toll booth in less than 3 minutes.