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This is a follow on to a similar question about 2 rolls of the same die. I'm not sure how to extend any of the numeric answers there to comparing two different sized dice if they're too large to make the one graphic solution feasible. ex a D61 and a D245

Dan Is Fiddling By Firelight
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1 Answers1

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Let's use a simple example, say a d4 versus a d7. Here's a chart from the link to the other question that you provided, adapted for the different dice:

\begin{array}{c|ccccccc} &1&2&3&4&5&6&7 \\ \hline 1&=&<&<&<&<&<&< \\ 2&>&=&<&<&<&<&< \\ 3&>&>&=&<&<&<&< \\ 4&>&>&>&=&<&<&< \end{array}

There are $4\cdot7 = 28$ possible outcomes, of which $4$ are draws, $6$ are when the d4 wins, and the rest when the d7 wins. Note that it is easier to calculate the number of "<" in the diagram than the ">", because the "<" form a triangle.

To generalise this, suppose that your dice have $m$ and $n$ sides, with $m < n$. Then there will be $m$ draws; the d$m$ will beat the d$n$ $m \choose 2$ times, and the rest of the time, the d$n$ will win.

In other words, the chance of a tie is $$\frac{m}{mn}=\frac1n,$$ the chance of the d$m$ winning is $$\frac{m \choose 2}{mn} = \frac{\frac{m(m-1)}2}{mn} = \frac{m-1}{2n},$$ and the chance of the d$n$ winning is $$1 - \frac1n - \frac{m-1}{2n} = \frac{2n-m-1}{2n}.$$

Théophile
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