The probability that some subset contains an odd number of elements.

[rain gazelle]

So, we want to find the sum of odd binomial coefficients.
We have:
(1 + x)^n = Σ(C(n, k)x^k, k = 1 to n)
(1 - x)^n = Σ(C(n, k)(-1)^k x^k, k = 1 to n)
Let's subtract them.
(1 + x)^n - (1 - x)^n = Σ(C(n, k)(1 - (-1)^k)x^k, k = 1 to n)
As 1 - (-1)^k = 1 - 1 = 0 for even k and 1 + 1 = 2 for odd k, we get:
(1 + x)^n - (1 - x)^n = 2Σ(C(n, k)x^k, k = 1 to n, k odd)
Thus:
Σ(C(n, k)x^k, k = 1 to n, k odd) = ((1 + x)^n - (1 - x)^n)/2
Substituting x = 1 gives:
Σ(C(n, k), k = 1 to n, k odd) = ((1 + 1)^n - (1 - 1)^n)/2 = 2^(n - 1)
So, the sum of odd-indexed binomial coefficient is just half the sum of them all. Kinda obvious when n is odd, but a bit less obvious when n is even, so quite useful.

Yes.