Probability of exactly 500 heads in 1000 flips

Question

You flip a fair coin 1000 times. Estimate the probability of getting exactly 500 heads. Derive the approximation, no calculator.

Accepted Answer

The exact answer is P = \binom{1000}{500}(1/2)^{1000}. Applying Stirling to the central coefficient with n = 500: P = \frac{\binom{2n}{n}}{2^{2n}} \approx \frac{1}{\sqrt{\pi n}} = \frac{1}{\sqrt{\pi \cdot 500}} = \frac{1}{\sqrt{1571}} \approx \frac{1}{39.6} \approx 0.0252. The exact value is 0.02523, so the estimate is spot on. Exactly half heads now happens only about 2.5\% of the time, less than a third as often as at 100 flips. That is the 	frac{1}{\sqrt{n}} law at work: quadrupling the flips (from 250 to 1000) halves the peak probability. Equivalently, the standard deviation grew to \sig…

Probability of exactly 500 heads in 1000 flips

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