# Probability estimation of minesweeper undecidable choices

Someone asked what was his probability to win in this situation?

This is an interesting problem and I'll be curious to know an efficient solution.

So far, I wanted also to generalize, with a simplification, we consider only:

• pairs of blocks such as the one on top right, and model them with a binary variable which value is the presence of the mine
• twin pairs, such as the one on bottom left, which is a couple of pairs linked together, the knowledge of one value of a pair gives the value of the other pair and vice versa

So we omit groups of more than 2 pairs linked together, for simplicity.

Now in this simple example, we have 2 single pairs and 2 twin pairs, let's denote twins with binary variables: $A_1, A_2$, and the other one with $B_1, B_2$ and the other single pairs with $C$ and $D$

So we could think the probability is $(\frac 1 2)^4$ but I'm not sure at all

The goal is to generalize with m twins pairs and n single pairs

• Do you know the total number of mines in the game or not? – whuber Jul 14 '14 at 18:59
• that's not needed, you know there's a mine in each pair of block, so with my notations, it remains 2m+n mines – caub Jul 14 '14 at 20:35
• Although it's not needed in your specific example, it can be needed in general settings. It sounds like you are stipulating that the total is known. – whuber Jul 14 '14 at 20:36
• the total in this situation near the end of the game yes, it is well known: 2m+n=number of remaining pairs=1/2*number of remaining blocks, sorry I wasn't clear – caub Jul 14 '14 at 20:46

You are correct with your answer of $\left(\frac{1}{2}\right)^4$. You have arrived at the unfortunate situation of having less than a 7% chance of winning!

Single and double pairs both have probability $\frac{1}{2}$ of being correctly solved assuming no mistakes are made after the first click. For single pairs this is obvious, for double pairs the first click will always either reveal a mine or a number, both of which provide enough information to complete the rest safely. This logic can be extended to any pattern of this sort, and is also true for triple pairs etc.

Each single/double pair is independent as long as they are not next to each other. As there is a fixed number of mines in each pair, the configuration has no effect on another pair, as long as they don't share any revealed numbers.

For a total of $n$ groups of pairs left at the end of a game, which may be single, double, triple etc., the probability of successfully completing the game assuming no mistakes are made is $\left(\frac{1}{2}\right)^n$.