Dice in the Middle
You and I each have dice: you have two six-sided dice, I have one. We roll all dice simultaneously. For me to win, my roll must be strictly between the numbers rolled by your dice. What is the probability that I win?
Answer
We condition on wether there are ties or not (if there are ties the conditional probability is zero).
We compute the probability that all three rolls are distinct (no ties):
\[
\frac{6}{6}\times\frac{5}{6}\times\frac{4}{6}
=\frac{5}{9}.
\]
Given no ties, each of the three rolls is equally likely to be the smallest, middle, or largest. Hence
\[
P(\text{middle}\mid\text{no ties})=\frac{1}{3}.
\]
Therefore
\[
P(\text{win})
=\frac{5}{9}\times\frac{1}{3}
=\frac{5}{27}.
\]