Probability of exactly 100 sixes in 600 rolls

Question

You roll a fair die 600 times and count how many sixes appear. Estimate the probability of getting exactly 100 sixes (the mean). Derive the approximation, no calculator.

Accepted Answer

The count of sixes is binomial with n = 600 and p = 	frac16, so the mean is np = 100, exactly the value we are asking about. The exact probability \binom{600}{100}(1/6)^{100}(5/6)^{500} is hopeless by hand, so use the local central limit theorem: near its peak a binomial looks Gaussian, and a Gaussian puts probability about 	frac{1}{\sigma\sqrt{2\pi}} on its central value. Compute the standard deviation: \sigma = \sqrt{n\,p\,(1-p)} = \sqrt{600 \cdot 	frac16 \cdot 	frac56} = \sqrt{	frac{3000}{36}} = \sqrt{83.3} \approx 9.13. Then P(	ext{exactly }100) \approx \frac{1}{\sigma\sqrt{2\pi}} =…

Probability of exactly 100 sixes in 600 rolls

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