The Minesweeper Page - Probability Calculation


Warning: There is mathematics on this page. So to remind you:

The binominal coefficient n over k= n! / (k! * (n-k)!) gives you the number of possibilities to distribute K mines on N squares. In our case, we take K=99 mines and N=30*16=480 squares for the expert mode.


In the situation:

there are two possible mine schemes:

Nr 1 :

In the three unopened squares of interest (the square in the corner does not matter) there is one mine. That means, the other k-1 mines of the global field can be distributed on the remaining n-3 squares. And this can be done in (n-3) over (k-1)different ways.

Nr 2 :

In the three squares there are two mines. The distribution of the remaining k-2 mines on the other n-3 squares can be arranged in (n-3) over (k-2)ways.


The possibility of scheme Nr. 2 is the number of possible arrangements for the scheme Nr. 2, divided by the number of arrangements for the situation, that means, divided by the sum of the numbers for scheme 1 and scheme 2:

p2 = (n-3) over (k-2)/ ((n-3) over (k-1) + (n-3) over (k-2))

To calculate this, we need the well known formula for these coefficients:

(n-3) over (k-1)=(n-3) over (k-2)* (n-3-(k-2)) / (k-1)

So we get:

p2 = 1 / (1 + (n-k-1) / (k-1)) = (k-1) / (n-2) = k/n

(The last equation assumes, that k and n are big numbers).

This means: The possibility of scheme Nr. 2 is nearly the possibility of encountering a mine when opening a random square.


Back to the Minesweeper Probability Page.

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