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Thread: need prob. help fast

  1. #1
    Jul 2007

    need prob. help fast

    Find the probability of rolling a total of 8 using 3 die.

    I want to do this without having to write down every option individually.

    Thanks for the help.
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  2. #2
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    The total number of combos is $\displaystyle 6^{3}=216$

    $\displaystyle \left(\sum_{k=1}^{6}x^{k}\right)^{3}=(x+x^{2}+x^{3 }+x^{4}+x^{5}+x^{6})^{3}$

    Can you expand this?. Look at the coefficient of the x term whose exponent is 8. That's the number of ways it's possible to sum to 8 with 3 dice.

    The expansion will start with x^3 and the first coefficient is 3. So, we have $\displaystyle x^{3}+3x^{4}$, thus far.

    Now, to find the next coefficient, multiply the coefficient of the x^4 term by the exponent 4 and divide by one less than the preceding exponent. (3)(4)/2=6

    $\displaystyle x^{3}+3x^{4}+6x^{5}$


    $\displaystyle x^{3}+3x^{4}+6x^{5}+10x^{6}$

    Proceed in that manner a few more times and we get $\displaystyle 21x^{8}$

    The probability of rolling a sum of 8 with 3 dice is $\displaystyle \frac{21}{216}=\frac{7}{72}$
    Last edited by galactus; Jul 13th 2007 at 05:27 PM.
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