# 2008 iTest Problems/Problem 21

## Problem

One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called Risk that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Michael explains a bit of strategy.

"You have the first move and you occupy three of the four territories in the Australian continent. You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn."

"Don't tell her that!" complains Joshua.

Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia. Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack. Finally, Wendy is left with one chance. Wendy and Joshua each roll just one six-sided die. Wendy wins if her roll is higher than Joshua's roll. Let a and b be relatively prime positive integers so that $a/b$ is the probability that Wendy rolls higher, giving her control over the continent of Australia. Find the value of $a+b$.

## Solution

The successful outcomes (shown in table where rows are Wendy’s rolls and columns are Joshua’s rolls) are straightforward to count — whatever Joshua rolls, find the numbers greater than his result (if any).

 1 2 3 4 5 6 1 2 $\cdot$ 3 $\cdot$ $\cdot$ 4 $\cdot$ $\cdot$ $\cdot$ 5 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 6 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$

The number of successful outcomes is $15$ and the total number of outcomes is $36$, so the probability that Wendy was able to get a higher roll (and get Indonesia and the Australian continent) is $\tfrac{5}{12}$. Thus, $a+b=\boxed{17}$.