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Math Help - Question on total probability formula

  1. #1
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    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
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  2. #2
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    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?
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  3. #3
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    Re: Question on total probability formula

    Hello, sakuraxkisu!

    I too have never heard of the total probability formula.
    Could you explain it?


    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}
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  4. #4
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    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
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