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.

The purpose of these lines is to adjust the look of this page. They are filling the surrounding table, so that the text begins directly right of the backgroundpicture. If you are reading this text, you are either doing strange copy /paste-actions, reading the HTML-Sourcecode or you are using a textbrowser. Whatever is right, you can simply ignore this paragraph.