what is the probability that 10 rolls of a fair die will show 6 fives?
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Assuming the question means exactly 6 (and no more than 6) fives, we can use the binomial expansion to find the probability of 6 fives and 4 non-fives. If p=1/6 is the probability of rolling a five then, 1-p=5/6. The binomial expansion of (1/6+5/6)^10=1 which means that all the possibilities of rolling 10 dice or one die 10 times sum to 1 or 100%. The expansion of the binomial uses the coefficients: 1 10 45 120 210 252 210 120 45 10 1. These coefficients are applied to the probability of 10 fives, 9 fives and one non-five, 8 fives and two non-fives, and so on. The 5th coefficient, 210, applies to the required condition. The probability is 210*(1/6)^6*(5/6)^4=0.002171, 0.2171% approx.

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