Question on total probability formula

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November 8th 2012, 10:11 AM
sakuraxkisu

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 :)
November 8th 2012, 08:18 PM
chiro

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?
November 8th 2012, 09:38 PM
Soroban

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: . ${64\choose2} \,=\,2016$ outcomes.

Consider placing a domino in a row: . $\boxtimes\!\!\boxtimes\!\!\square\!\square\! \square\! \square\!\square\! \square$
There are 7 possible positions.
On the entire board, there are $8\times 7 \,=\,56$ ways
. . to place a domino horizontally.

Consider placing a domino in a column.
There are 7 possible positions.
On the entire board, there are $8\times7 \,=\,56$ ways
. . to place a domino vertically.

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

The probability is; . $\frac{112}{2016} \:=\:\frac{1}{18}$
November 9th 2012, 09:38 AM
sakuraxkisu

Re: Question on total probability formula

In my lecture notes, it says that the formula is:
$\ P(A) = \sum_{i=1}^{K}\ P(A \cap B_{i}) = \sum_{i=1}^{K}\ P(A \mid B_{i}) P(B_{i})$
Assuming that $B_{1} , B_{2}, ..., B_{K}$ form a partition of the sample space and that $A \cap B_{1}, A \cap B_{2},..., A \cap B_{K}$ 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 :)