the mean is 4.5
probability that a call will take more than 3 minutes is equal to 80%
probability that a call will take less than 5 minutes is equal to 60%.
the probability that the call will take less than 4.5 minutes is equal to 50% because 4.5 is the mean.
the probability that the call will take more than 4.5 minutes is equal to 50% as well for the same reason.
the probability that the call will take between 4.5 and 5.0 minutes is equal to the probability that the call will take less than 5 minutes minus the probability that the call will take less than 4.5 minutes.
that becomes 60% minus 50% which is equal to 10%.
what is the probability that a call will take more than 6 minutes?
3 is the same distance from 4.5 as 6 is.
the normal distribution is symmetric.
we can use the probabilities for 3 to calculate the probabilities for 6.
here's how it works.
the probability that the call will take more than 3 minutes is equal to 80%.
this means that the probability that the call will take less than 3 minutes is 20%.
since 6 is the same distance from the mean as 3 is, then we get:
the probability that the call will take less than 6 minutes is 80%.
the probability that the call will take more than 6 minutes is 20%.
that's the symmetry at work as you will be able to see in the picture at the end of this presentation.
the probability that the call will take between 3 and 6 minutes is calculated as follows:
the probability that the call will take less than 6 minutes is equal to 80%.
the probability that the call will take less than 3 minutes is equal to 20%.
the probability that the call will take between 3 and 6 minutes is 80% minus 20% which is equal to 60%.
your last question.
the probability that a call will last exactly 4.5 minutes.
this is actually equal to 0%.
this is because the normal distribution is a continuous distribution and the probability of getting an exact value is theoretically 0.
you can find the probability of the difference between 2 very close numbers, like 4.499999999999 and 5.00000000001.
as the interval gets smaller and smaller, the width of the interval gets smaller and smaller, until the width of the interval becomes 0 at which point in time the probability becomes 0.
if the distribution was a discrete distribution, then you could find the probability of exactly a number, but not if the distribution is continuous.
a pictures of the distributions is shown below:
