What's the probability of rolling doubles on 2d6?Responses generally look like this:

16%or this:

anydice.comI find both responses deficient. The first simply gives an answer without teaching anything. (It is also incorrect. The probability is $1/6 = 0.1\overline{6}\approx17\%$. We'll calculate this later.) If a second question arises, our gamer is back in the same situation as before. Also, if the question is related to a game design, if the answer isn't satisfactory, the designer has no knowledge to help change the design for the better. In fact, without a proper context and understanding of the mathematics, it may even be hard to know what questions need to be asked.

The second response seems somewhat better (although it may seem flippant or patronizing), as it gives our gamer a tool to answer not only this question, but any question of a similar nature. However, I think it's an even worse answer. Tools like anydice are fine as far as they go, but without a proper theoretical foundation they can be useless or even dangerous. While using the tool, apparent patterns will inevitably manifest. These may or may not be real since, as we hopefully all know, correlation is not causation.

The responses above are examples of generosity towards fellow gamers, for which we should be grateful. They offer a fish; my goal is to help to learn to fish. With a proper basis in the appropriate mathematical theory, a gamer should be able to

- ask the right questions,
- answer them independently,
- know what the results mean, and
- have some idea of what to do if the results aren't satisfactory.

- make good assumptions and approximations and
- determine the limitations of an analysis due to the assumptions made.

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