Mathematically, the odds of getting exactly N heads after flipping a fair coin 2N times is surprisingly high - it asymptotically approaches 1/sqrt(pi * N) as N

Mathematically, the odds of getting exactly N heads after flipping a fair coin 2N times is surprisingly high - it asymptotically approaches 1/sqrt(pi * N) as N becomes large. For the problem in IJ, this comes out to 1/(3.14 * 54) which is 0.076776 - not far from the exact value of 0.076599. As another example, if you flip a fair coin 1000 times, you’ll get exactly 500 heads about 2.5% of the time. The proof is also similarly easy: it’s a straightforward application of Stirling’s approximation to the formula (2N!)/(N!*N!)/2^(2N).