If a fair coin is flipped until a head appears, what is the probability it takes an even number of flips?

If a fair coin is flipped until a head appears, what is the probability it takes an even number of flips?

Approach: Sum the probabilities for k = 2, 4, 6...

1/3. Sum of (1/2)^k for even k: 1/4 + 1/16 + 1/64... = (1/4) / (1 - 1/4) = 1/3.

Follow-up: What is the probability it takes an odd number of flips?

Key concepts: geometric series, summation.