It is 9:00 p.m. The time until Joe receives his next text message has an exponential distribution with mean 5 minutes. A text has not arrived for 5 minutes. Find the probability that none will arrive for 7 minutes.
I don't know where to start with, is it like P(X>7)? But then I don't know where to put in the 5 minutes. Please help.
On
Following Andre's interpretation, the answer is exp(-2/5) = 0.67032. Because this in under a (simulation) tag, here is a verification in R (for which the accuracy of the simulation happens to be a little better than expected):
> x = rexp(10^7, 1/5); mean(x[x>5] > 7)
[1] 0.670339
Simulation under the alternate interpretation would have to begin at time 8:55pm.
Hint: $\mathsf P(X>7|X>-5)=\mathsf P(X>7) =\ldots$
That is, because an exponential distribution is memoryless, the fact that no message has arrived for the previous five minutes is irrelevant to calculating the probability that none will arrive within the next seven.