how to find probability of exponential distribution question with a given mean size?

Question

A program reads and processes text files, which sizes are not known in advance. The program fails if the size of the file is larger than$ 20$ GB. If the size of the file is modeled with an exponential distribution with mean size of $5 $GB. What is the probability that the program fails at least twice when it is tested with$ 100 $files?

Answer 1

Recall that the one-dimensional exponential distribution for a mean $\mu$ is proportional to $p(x) \sim e^{x/\mu}$, where we'll ignore normalization constants.

The probability a given file fails is:

${\int\limits_{x=100}^\infty e^{-x/5}\ dx \over \int\limits_{x=0}^\infty e^{-x/5}\ dx} = {1 \over e^{20}} \equiv p$.

The probability no files fail is:

$p_0 = {100 \choose 0}p^0 (1-p)^{100}$.

The probability that exactly one file fails is given by a binomial:

$p_1 = {100 \choose 1} p^1 (1-p)^{99}$

And the probability two or more fail is $1 - p_0 - p_1$