I'm struggling with this exponential distribution question.

Question

I have been able to get that the shower will last for 10 minutes or more with the probability of 0.135. However I am not sure what to do with the second part of the question. The question is as follows:

The duration (in minutes) of showers on a tropical island is approximately exponentially distributed with a rate = 1/5 [ X ~ E(1/5) ] .

Out of 3 showers, what is the probability that not more than 2 will last for 10 minutes or more?

Thanks for the help!

Answer 1

Inspect the complement of your set which is "all showers will last for 10 minutes or more" then the probability you're looking for is

1 - P("all showers will last for 10 minutes or more")

Make the natural assumption that all showers last independently, then it's an easy calculation.

Answer 2

If p is the probability that the rain lasts 10 minutes or more then (assuming independence):

P(not more than two showers last for 10 minutes or more)

$=1$-P(all three showers last for 10 minutes or more)

$=1-p^3$