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Math Help - Six-Sided Dice Probability - Getting a 5 or 6

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    Six-Sided Dice Probability - Getting a 5 or 6

    What are the chances of four of X (variable) rolled six-sided dice producing a result of 5 or 6 in one roll?
    4 dice = 1.2%
    5 dice = ???
    6 dice = ???
    etc.

    I know that each die has a 1 out of 3 (or 1/3, or 33.4%) chance of producing the desired result of 5 or 6 (also called a "hit"). I know that the chances of all four dice of a four-die roll "hitting" is equal to the chance of one of them hitting times the chance of each other one hitting, i.e.
    (1/3)*(1/3)*(1/3)*(1/3)

    Thank you for your time and your help!
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    Re: Six-Sided Dice Probability - Getting a 5 or 6

    Quote Originally Posted by jpmadigan1 View Post
    What are the chances of four of X (variable) rolled six-sided dice producing a result of 5 or 6 in one roll?
    4 dice = 1.2%
    5 dice = ???
    6 dice = ???
    etc.!
    I simply do not understand.

    Are you asking: If we roll seven dice, what is the probably getting at least one 5 or 6?

    OR, Are you asking: If we roll seven dice, what is the probably getting at exactly four 5's or 6's?

    Or is it some else all together?
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    Re: Six-Sided Dice Probability - Getting a 5 or 6

    If we roll seven dice, what is the probability of getting exactly four 5's or 6's.

    Sorry for the ambiguity.
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    Re: Six-Sided Dice Probability - Getting a 5 or 6

    Quote Originally Posted by jpmadigan1 View Post
    If we roll seven dice, what is the probability of getting exactly four 5's or 6's.
    Sorry for the ambiguity.

    If N\ge 4 is the number are dice rolled the probability of getting exactly four 5's or 6's is:
    \binom{N}{4}\left(\frac{2}{3}\right)^4\left(\frac{  1}{3}\right)^{N-4}

    where \binom{N}{4}=\frac{N!}{4!(N-4)!}
    Last edited by Plato; January 18th 2013 at 01:36 PM.
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