Question 515234: Nite Time Inn has a toll free telephone number so that customers can call at any time to make a reservation. A typical call takes about 4 minutes to complete, and the time required follows an exponential distribution. Find the probability that :
(a) a call takes 3 minutes or less
(b) a call takes 4 minutes or less
(c) a call takes 5 minutes or less
(d) a call takes longer than 5 minutes
Answer by drcole(72)   (Show Source):
You can put this solution on YOUR website! An exponential distribution with parameter  tell us the probability that some process will last a given amount of time or less, given that we know that the mean length of such a process is  . In this case, we know that a typical call takes about 4 minutes to complete, so
 (taking the reciprocal of both sides)
Given , the probability that a process will last T minutes or less is given by the cumulative distribution function:
In this case,  , so our function becomes:
Now let's answer your questions:
a) The probability that a call takes 3 minutes or less is:
 (to the nearest thousandth)
This tells us that it is more likely than not that a call will last three minutes or less, even though a typical call is four minutes long. This result is not as counterintuitive as it looks: longer calls are somewhat rare, but when they happen, their lengths push the typical call length higher than the median call length (the median call length is the call length for which half of all calls are shorter, and half are longer).
b) The probability that a call takes 4 minutes or less is:
 (to the nearest thousandth)
c) The probability that a call takes 5 minutes or less is:
 (to the nearest thousandth)
d) A call length is either less than or equal to 5 minutes, or greater than 5 minutes. So the sum of the probability that a call takes 5 minutes or less and the probability that a call takes more than 5 minutes should be 1:
 (subtracting  from both sides)
 (substituting using our answer from part (c))
 (distributing the negative)
 (simplifying)
 (to the nearest thousandth)
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