Probability of exactly 30 heads in 60 flips

Question

You flip a fair coin 60 times. Estimate the probability of getting exactly 30 heads. Derive the approximation, no calculator.

Accepted Answer

The exact answer is P = \binom{60}{30}\left(\frac{1}{2}\right)^{60}. Stirling applied to the central binomial coefficient (with n = 30) gives the standard shortcut \binom{2n}{n} \approx \frac{4^n}{\sqrt{\pi n}} \quad\Longrightarrow\quad P = \frac{\binom{2n}{n}}{2^{2n}} \approx \frac{1}{\sqrt{\pi n}}. So P \approx \frac{1}{\sqrt{\pi \cdot 30}} = \frac{1}{\sqrt{94.2}} \approx \frac{1}{9.71} \approx 0.103. The exact value is 0.1026, so the estimate is good to three decimals. Exactly half heads happens only about 10\% of the time here: the most likely single count, yet still uncommon, because the…

Probability of exactly 30 heads in 60 flips

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