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# Explaining probability

The mean time between reoccurring events is exponentially distributed with a mean time between events of 15 minutes.

a. What is the probability that there are no events within a 30-minute period? Answer 0.00043 ?
b. What is the probability that at least one call arrives within a 10-minute interval? Answer 0.4866 ?
c. What is the probability that the first events occurs within 5 and 10 minutes after the start of the sampling period? Answer 0.2031 ?
d. Determine the length of an interval of time such that the probability of at least one event in the interval is 0.90. Answer 34.54 ?

#### Solution Preview

a. What is the probability that there are no events within a 30-minute period? Answer 0.00043 ?
Mean time between events = 15 minutes
Average number of events in 30 minute period = 30/15=2
This is a poisson distribution with lambda=2 and x=0
P(x=0) = 2^0*e^-2/0!
=1/1*e^2 = 0.1353

b. What is the probability that at least one call arrives within a 10-minute interval? Answer 0.4866 ?
Mean time between events = 15 minutes
Average number of events in 10 minute period = 10/15=2/3
This is a poisson distribution with lambda=2/3
Now
P(x=0) = ...

#### Solution Summary

The mean time between reoccurring events is exponentially distributed with a mean time between events of 15 minutes.

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