The case of particular interest for gamers with respect to the "law of large numbers" is:
Let the random variable Y be equal to the number of successful dice rolls throughout n independent dice rolls with probability p of success. (ie. This follows a binomial distribution with n trials and probability p of success). The ratio Y/n is the relative frequency of success.
One can show via the binomial distribution and Tchebyshev's inequality, that for every fixed epsilon > 0,
limit {n->infinity} Probability( |Y/n - p| < epsilon) = 1
(Note that this is only true in the limit when the number of independent dice rolls approaches infinity. It says very little about the case where the number of independent dice rolls is finite).
Let the random variable Y be equal to the number of successful dice rolls throughout n independent dice rolls with probability p of success. (ie. This follows a binomial distribution with n trials and probability p of success). The ratio Y/n is the relative frequency of success.
One can show via the binomial distribution and Tchebyshev's inequality, that for every fixed epsilon > 0,
limit {n->infinity} Probability( |Y/n - p| < epsilon) = 1
(Note that this is only true in the limit when the number of independent dice rolls approaches infinity. It says very little about the case where the number of independent dice rolls is finite).
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