I use the internet for two hours each day.
What is the probability that when I go to my computer to use the internet, I find mt connection is down?
Assuming that my internet goes down randomly any time of the day or night, and that I want to use the internet with equal probability any time of the day or night, then both the time at which the internet connection goes down and the time at which I want to use the internet have a uniform distribution.
\[P(internet down) = \frac{1}{24}\]
\[P(I want to use internet) = \frac{2}{24}= \frac{1}{12}\]
I do not know whether or not the internet is working until I try to use it, so I can assume these two are independent,
Then
\[\begin{equation} \begin{aligned} P(internet \: down | I \: want \: to \: use \: it) &= \frac{P(I \: want \: to \: use \: the \: internet \: and \: find \: it \: is \: down)}{P(I \: want \: to \: us \: the \: internet)} \\ &= \frac{P(I \: want \: to \: use \: the \: internet) \times P(The \: internet \: is \: down)}{ P(The \: internet \: is \: down)} \\ &=P(I \: want \: to \: use \: the \: internet) =\frac{1}{12} \end{aligned} \end{equation}\]