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Math Help - Probability problem with 4 Dice

  1. #1
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    Probability problem with 4 Dice

    I'm stuck with the following problem: If I roll 4 regular six-sided dice what is the probability that the total will be greater than or equal to 20? I can't work out how to calculate this short of writing out all the combinations of throws that would yield a sum of at least 20. Ideally I would like to derive a formula.
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  2. #2
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    You could convolutions twice, which would be a bit easier (still messy though!)

    The first convolution gets the distribution of the sum of a pair of dice.

    Then do convolutions of that distribution to see the distribution of two [I]pairs of[I] dice.


    If you want to check your answer, google suggests:
    http://www.stat.science.cmu.ac.th/~w...0_132_P001.pdf
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  3. #3
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    Quote Originally Posted by StaryNight View Post
    I'm stuck with the following problem: If I roll 4 regular six-sided dice what is the probability that the total will be greater than or equal to 20? I can't work out how to calculate this short of writing out all the combinations of throws that would yield a sum of at least 20. Ideally I would like to derive a formula.
    Go to this page
    You will see the expansion of \left( {\sum\limits_{k =1 }^6 {x^k } } \right)^4 .
    The coefficients of the x^n tell you how many ways one can roll a sum of n with four dice.
    Now how can you use that to work this question?
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  4. #4
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    Quote Originally Posted by Plato View Post
    Go to this page
    You will see the expansion of \left( {\sum\limits_{k =1 }^6 {x^k } } \right)^4 .
    The coefficients of the x^n tell you how many ways one can roll a sum of n with four dice.
    Now how can you use that to work this question?

    that is the most interesting bit of probability/statistics ive ever seen
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  5. #5
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    Quote Originally Posted by Plato View Post
    Go to this page
    You will see the expansion of \left( {\sum\limits_{k =1 }^6 {x^k } } \right)^4 .
    The coefficients of the x^n tell you how many ways one can roll a sum of n with four dice.
    Now how can you use that to work this question?
    I can sum the number of ways of getting 20,21,22,23,24. But why does the formula you gave work?
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  6. #6
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    Quote Originally Posted by StaryNight View Post
    But why does the formula you gave work?
    That is known as a generating function. It is a standard technique used in general counting problems such as this problem. It is a very difficult problem to do by hand.

    I fail to understand why you are expected to solve a problem that you do not have the tools to use. Maybe that is a conversation you can have with your education authority.
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  7. #7
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    We've covered probability generating functions, but these were used to calculate E(X), E(X^2) etc.. I think this is different.
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    Quote Originally Posted by StaryNight View Post
    We've covered probability generating functions, but these were used to calculate E(X), E(X^2) etc.. I think this is different.
    It is somewhat different.
    Here is a free download of a standard textbook..
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  9. #9
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    Quote Originally Posted by Plato View Post
    Having reread my notes on PGF's, I can see that the PGF for a sum of random variables is equivilent to the PGF's of those random variables multiplied together. Using the PGF for a uniform distribution, this gives the equation you suggested using. What I am wondering is how one calculates the expansion without a computer in reasonable time - this problem was actually set as part of an exam question. - http://www.cl.cam.ac.uk/teaching/exa.../y2001p2q5.pdf.
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