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Stephen Tashi
Feb17-12, 03:01 PM
Sci Advisor
P: 3,314
Abstractly you have 3 events:
E = there is a mine in the square with the question mark
A = there is a 3 in the square you showed
B = there is a 2 in the other square you showed.

You have stated the "conditional probabilities"
P(E given A) = 2/3
P(E given B) = 1/2

The probability you are asking about is
P( E given (A and B) )

As far as terminology goes, I don't know any special name for this probability. (Minesweeper enthusiasts may have a name for it that I don't know about.) For an arbitrary set of events E,A,B, there is no general formula for calculating this probability using only the values of P(E given A) and P(E given B). You can write various expressions to calculate P(E given (A and B)) but they would include terms like P( E and A and B) and P(A and B). In the particular situation you gave, you can calculate P(E given (A and B)) by a combinatorial analysis.