An exceedingly regular coin

If I throw a fair coin (one that always lands on a face, never on its edge, and has equal probability of showing heads or tails) one hundred times, I know that it is not guaranteed to give exactly fifty heads and fifty tails, of course. However, exactly fifty heads and fifty tails (in any order) is certainly possible. What are the chances of it happening?

If the coin is thrown 2n times, then the number of distinct outcomes (each being a distinct sequence of 2n occurrences of heads or tails) is 2^2n (where ^ means power), because each occurrence has two possible states (heads or tails) and there are 2n such occurrences in total.

If an outcome (being a sequence of 2n occurrences of heads or tails) comprises h heads and t tails, with h+t=2n, then the number of such outcomes is (2n)!/(h!t!), because the number of ways of arranging 2n occurrences is 2n! but each such result comprises h occurrences of heads, which can be arranged in h! ways, and t occurrences of tails, which can be arranged in t! ways, so we have counted each distinct result h!t! times. If we are looking for cases where h=n and t=n, then the number of such outcomes is (2n)!/(n!n!), or (2n!)/(n!)^2.

So, the proportion of possible outcomes from 2n throws which comprise exactly n heads and n tails is (2n!)/((n!)^2)(2^2n), or (2n!)/(n!(2^n))^2 (because 2^2n is (2^n)^2).

This gives us the following results:

• For 2 throws, the probability of 1 heads and 1 tails is 0.5, or 1 in 2 (which is pretty obvious, because the outcomes are HH, HT, TH, TT, and 2 of those 4 outcomes comprise 1 head and 1 tail).
• For 4 throws, the probability of 2 heads and 2 tails is 0.375, or 1 in 2.666...
• For 6 throws, the probability of 3 heads and 3 tails is 0.3125, or 1 in 3.2
• For 8 throws, the probability of 4 heads and 4 tails is 0.2734375, or 1 in 3.657...
• For 10 throws, the probability of 5 heads and 5 tails is 0.24609375, or 1 in 4.063...

To answer our original question:

• For 100 throws, the probability of 50 heads and 50 tails is 0.079..., or 1 in 12.564...

Looking further ahead:

• For 1000 throws, the probability of 500 heads and 500 tails is 0.025..., or 1 in 39.643...

So there is a little more than a one in thirteen chance of getting exactly fifty heads and fifty tails in one hundred throws of the coin. Far from a dead cert, it is true, but more likely than I would have expected.

Further reading

The Coin Flip: A Fundamentally Unfair Proposition?