First the problem, then the 2 solutions I have. The question: How do I know which method is correct? ie, test for independence and how would I have known to do that?

Players A, B, and C take turns flipping a fair coin. The first to flip heads (1) wins. They flip in the order presented; A, B, C.

-Describe the sample space

-What is P(A wins)?

Sample space:

S = {1, 01, 001, 0001, ... , 000000...} where the game ends if a heads (1) is flipped.

Method 1:

A has to flip heads on the 1 mod 3 flip to win the game. However, if A is flipping, this means nobody has won the game. Every time A flips, we disregard all prior flips. Thus, every 1 mod 3 flip, P(A wins) = 1/2.

Method 2 (infinite bernouli, I think):

We sum the infinite converging series 1/2 + 1/8 + 1/64 + ... + 1/[2^(3k+1)] where k = 0,1,2,... representing each round/iteration, 3k+1 is A's kth flip. Thus, P(A wins) = 4/7. This math can be done for B and C as well, with all of their probabilities adding to 1.