Using Bayes' Theorem to Track Probability Distribution of Minesweeper Grid (Part 2)

This post is part of a series, please see the first post for more background.

A successful attempt by the solver

Developing the Solver

When a game of minesweeper is begun, two pieces of information are known: the number of elements in the grid and the number of mines. From this information, the probability of containing a mine for each element in the grid is

This seems intuitive, but it is actually a special case of a more general form.

The probability of a random event can be expressed as the number of outcomes for which the event occurs divided by the total number of possible outcomes. In the case of minesweeper, the probability of a  particular square containing a mine can be found by dividing the number of configurations of mines that include the square by the number of possible configurations.

The example below illustrates the probability of a mine occupying the square in the top left corner.

Three favorable configurations

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Six possible configurations

The number of possible configurations of mines has a special name and notation. The expression in parentheses below is pronounced "n choose k". It is equal to the number of distinct k-sized subsets that can be selected from a set of n elements.

Binomial Coefficient

Consider the animations above. The number of favorable configurations at the top would be calculated as "three choose one". Meanwhile, on the bottom, the number of possible configurations is equal to "four choose two". For the purpose of calculating the initial probabilities at the start of the game, they should be equal across the board. For each element, dividing the number of favorable configurations by the total number of configurations  reduces to the intuitive ratio that we started with.

This seems like a lot of unnecessary thought went in to calculating something as intuitive as the initial distribution. However, as the grid becomes populated with measurements, we will have to augment our approach to calculate the new distributions.