The Minesweeper Page - About Probabilities

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Sometimes you encounter a situation, where you cannot exactly say, where a mine is. But maybe some mine groupings occur more often than others. If you know, where the mine probably is, you will not loose as often as if you guessed. In this chapter I try to show you how to get a chance of guessing, where no mine is.


Every few games you will encounter the following situation:

There are two possible mine arrangements for this situation:

Nr 1 :

Nr 2 :

There are two ways to solve this with a minimum risk:

The first way is to open the square in the corner. The probability of hitting a mine is approximately 1/5 (99 mines in 16*30=480 squares, 99/480=0.20625=20.625%=1/5), that means, you will hit a mine every fifth try. But if there is a "1" under the square, pattern 1 is correct; if there is a "2", scheme 2 is correct.

The second way is a bit more simple: just assume, that pattern 1 is correct. The probability of this scheme failing is also 1/5. So you can mark a mine to the side of the "2"'s and continue with left-right-clicking on the "1"'s and the squares that will open. Using this way, you don't need to wait until you recognize, what number is below the corner square.

Maybe you are interested, how to calculate the probability of both groupings.


The next situation won't occur so often, but will demonstrate a certain minesweeper principle:

You may open the square directly right to the lower mine, without risking a hit. If you see a "1" popping out:

all is OK, but if you see a "2" under the opened square:

you will have to guess.

There are three possible situations, from which you have to decide for one:

Nr 1 :

Nr 2 :

Nr 3 :

In this case, the most probable situation is Nr 1. It has a probability of approximately 4/5 (exactly 100%-22,89%=77,11%, so it's a bit different from the situation at the top of the page).


The last situation showed you a characteristic feature of minesweeper: One mine pattern is more possible than another, if there are less mines in it. This is so, because there is about 1 mine every 5 squares, so it is more possible to have only 1 mine in e.g. three squares than to have 2 mines in these squares. With other words: if you don't know where to set a mine, set it at a place, where more numbers share a mine.

So, if you had to guess in this situation:

you would open the following squares:

and hope, you can continue without guessing more. In this situation, it is more possible, that there is one mine for both "2"'s than that each "2" would have it's own mine at the squares which you should open.


A secret programmers conspiration? - or: Is Minesweepers Random Number Generator OK?

Did you ever start Minesweeper (Windows version) with a user defined size of 30*24 squares and only 10 mines. Did you assume, that you get a 1-second-time. Wrong. Most of your tries, you have to open another square. Why?

A little calculation: There are 30*24 squares=720 squares for 10 mines, that means, one mine owns a room of 72 squares=8.5 * 8.5 squares. And that means, between two mines should be a space of 8.5 squares. So why doesn't the whole field pop open every time?

One answer would be: Minesweepers random number generator does not function correct. Do the minesweeper programmers want, that we do not believe in probabilities any longer? In this case, the thoughts on this page about where to put a mine weren't correct.

Another answer is: This little 8.5-squares-calculation is not correct. Maybe there is a better calculation and Minesweeper should not open every time. This is true. Maybe we are angered too often by a damned false guess, but that's not because there is a fault in minesweeper, but because we play minesweeper too often.


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