Rooting For The Underdog Die

Relevant wiki: Probability - Problem Solving

If you roll a $1$ , then your opponent has $1$ possible roll that would allow you to win (If he rolled a $1$ ).

If you roll a $2$ , then your opponent has $2$ possible rolls that would allow you to win (If he rolled a $1$ or a $2$ ).

If you roll a $3$ , then your opponent has $3$ possible rolls that would allow you to win.

This pattern continues for all of your $6$ possible rolls. Each of your rolls has a $\large\frac{1}{6}$ probability. Meanwhile, each of your opponent's rolls has a $\large\frac{1}{8}$ probability.

Your roll and your opponent's rolls are independent , and so we calculate the probability of winning by multiplying the probability of your roll by the probability of your opponent's roll.

$P(\text{you roll a 1}) \cap P(\text{your opponent rolls a 1})={\large\frac{1}{6}}\times{\large\frac{1}{8}}$

$P(\text{you roll a 2}) \cap P(\text{your opponent rolls a 2 or less})={\large\frac{1}{6}}\times{\large\frac{2}{8}}$

$P(\text{you roll a 3}) \cap P(\text{your opponent rolls a 3 or less})={\large\frac{1}{6}}\times{\large\frac{3}{8}}$

$\ldots\text{and so on}$

These probabilities are mutually exclusive, and so we can use the Rule of Sum to find $P(\text{you win})$ .

Summing the probabilities yields $P(\text{you win})={\large\frac{7}{16}}$ , and so $a+b=\boxed{23}$ .

If you roll $n$ , you have an $\frac{n}{8}$ chance of winning.

So, the probability of winning is

$P(\text{You Win})=\frac{1}{6}(\frac{6}{8} + \frac{5}{8} + \frac{4}{8} + \frac{3}{8} + \frac{2}{8} + \frac{1}{8})=\frac{7}{16}$

Since $7$ and $16$ are coprime, their sum, $23$ is the solution.