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Given a 12 sided fair die write down a probability function that gives the probability of rolling x '7's from 20 rolls.

I'm not really sure where to start on this. But I know there are 12^20 total possibilities and that each time you roll the die there is a one in 12 chance of getting a 7. If I did (1/12)^20 that would give me the probability of rolling 20 7s in a row. But how do I include 7s that aren't rolled in a row?

Any ideas would be appreciated very much!

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In a rudimentary manner, favorable ways $=\dbinom{20}{x}\cdot 1^x\cdot 11^{20-x}$, against $12^{20}$ total ways,

or you could use the familiar binomial distribution,

$P(X = x) = \dbinom{20}{x}\cdot\left(\dfrac{1}{12}\right)^x\cdot\left(\dfrac{11}{12}\right)^{20-x}$

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The answer is $\binom{20}{x}12^{-20}$ since any dice pattern is equally likely and all you need is to count the number of possible patterns of x 7's out of 20 and divide by the total number of possibilities.

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  • $\begingroup$ The answer would be correct only if p = q = 1/2. Here p = 1/12, q = 11/12. $\endgroup$ Oct 19 '15 at 17:59
  • $\begingroup$ of course you're right! $\endgroup$
    – kodlu
    Oct 19 '15 at 21:39

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