Show that given n coins, if the probability of getting heads an even number of times is 1/2 then there is at least one fair coin

Not as Elegant a Basic Approach as I Had Foreseen. We show this by induction.

First, for $n = 1$, a single coin. Obviously, then the probability of an even number of heads is simply the probability that this coin flips tails. If this coin is unfair, this probability is clearly not equal to $1/2$. Therefore the coin must be fair. This establishes the basis step.

Now, suppose that the proposition is true for some $n > 0$. Let us now show it for $n+1$. The antecedent is that the probability of an even number of heads in these $n+1$ flips is $1/2$. If (at least) one of the first $n$ coins is fair, then the consequent is true.

If, on the other hand, none of the first $n$ coins is fair, we already know that such a circumstance does not permit the probability of an even number of heads in the first $n$ tosses to be $1/2$. Let us say therefore that this probability is instead $P_n \not= 1/2$, and let the $n+1$th coin have a probability of heads of $q$. Then the probability that the number of heads is even after all $n+1$ tosses is

$$ P_{n+1} = P_n(1-q) + (1-P_n)q = P_n + q(1-2P_n) $$

But we know, by hypothesis, that $P_{n+1} = 1/2$, so we write

$$ \frac12 = P_n + q(1-2P_n) $$

which gives us, after some simple algebra,

$$ q = \frac{1/2-P_n}{1-2P_n} = \frac12 $$

This establishes the induction step and the proposition is shown.