Question on total probability formula

Hi, my question is:

Two different squares are selected at random on an 8 x 8 chessboard. What is the probability that they share a common boundary (i.e. that htey have an edge in common, not just a single corner)?

For this question, it says I'm supposed to use the total probability formula. Any help would be greatly appreciated :)

Re: Question on total probability formula

Hey sakuraxkisu.

I'm not familiar with that term, but you could point out what the total probability formula is in your book/lecturer/whatever?

Re: Question on total probability formula

Hello, sakuraxkisu!

I too have never heard of the total probability formula.

Could you explain it?

Quote:

Two different squares are selected at random on an 8 x 8 chessboard.

What is the probability that they share a common boundary?

Selecting 2 of the 64 squares, there are: .$\displaystyle {64\choose2} \,=\,2016$ outcomes.

Consider placing a domino in a row: .$\displaystyle \boxtimes\!\!\boxtimes\!\!\square\!\square\! \square\! \square\!\square\! \square$

There are 7 possible positions.

On the entire board, there are $\displaystyle 8\times 7 \,=\,56$ ways

. . to place a domino horizontally.

Consider placing a domino in a column.

There are 7 possible positions.

On the entire board, there are $\displaystyle 8\times7 \,=\,56$ ways

. . to place a domino vertically.

Hence, there are $\displaystyle 56+56\,=\,112$ pairs of adjacent squares.

The probability is; .$\displaystyle \frac{112}{2016} \:=\:\frac{1}{18}$

Re: Question on total probability formula

In my lecture notes, it says that the formula is:

$\displaystyle \ P(A) = \sum_{i=1}^{K}\ P(A \cap B_{i}) = \sum_{i=1}^{K}\ P(A \mid B_{i}) P(B_{i}) $

Assuming that $\displaystyle B_{1} , B_{2}, ..., B_{K} $ form a partition of the sample space and that $\displaystyle A \cap B_{1}, A \cap B_{2},..., A \cap B_{K} $ form a partition of A.

I got the same answer as you Soroban, although my method was slightly different. I think what confused me most was how this formula could be used in the question. Thank you anyway though :)