When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket. What is the probability that the bus is filled in less than 3 hours from the time of the fare reduction

The probability that the bus is filled in less than 3 hours from the time of the fare reduction is 0.9975.

Step-by-step explanation:

We are given that the time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket.

Let X = the time between calls for tickets

The probability distribution for the exponential distribution is given by;

,  where x > 0.

Here, = exponential parameter.

As we are given in the question that the mean of the exponential distribution is 30 minutes, i.e;

Mean of an exponential distribution =  

For calculating the required probability, we have to use the cumulative distribution function of an exponential distribution which is given by;

;  where x > 0

Now, the probability that the bus is filled in less than 3 hours from the time of the fare reduction is given by = P(X < 180 minutes)

          P(X < 180 min) =  

                                   =  

                                   =  0.9975

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