I’m thinking about a sequence of coin flips resulting in either heads or tails, with equal probability. Which of the following statements are true?

In a game where I win if we flip HTH on consecutive throws and my friend wins if we flip HTT on consecutive throws:

1) If we just toss the coin three times in succession, we each have an equal probability of winning (although most of the time neither of us would win).

2) If we keep flipping until either HTH or HTT comes up, we will win the game with equal probability.

3) If we keep flipping until either HTH or HTT comes up, the average number of flips it will take is on average the same, whether I win or my friend does.

4) If we each have our own coin, and I keep flipping until I see HTH on consecutive flips, and my friend keeps flipping her coin until she sees HTT on consecutive flips, we will both take the same number of flips on average.