Question

Two dice, one red and one white are rolled. What is the probability that the white die turns up a smaller number than the red die ?

Answer 1

You can do this a few ways (and there's even a cached answer from this site that helps quite a lot as well) but I found this answer while looking around and think it's super cool.

First and easiest, if you make a list and count all of the possible true outcomes, it won't take that long. You know that the 6 1's will never have a smaller roll and all but one of the 6's will....

But the neat way is this:

normal 6 sided die probability for 1 roll
\[P _{(1roll)}= 1/6\]

\[P _{(Red>White)}+P _{(Red<White)}+P _{(Red=White)}=1\]
and symmetry says that
\[P _{(Red>White)}=P _{(Red<White)}\]
so

\[2P _{(Red>White)}+P _{(Red=White)}=1\]

...

Answer 2

http://openstudy.com/study#/updates/4eec24c7e4b0367162f56955