Statements 1, 2 and 3 are true.

Statement 1: HTH and HTT will each happen 1/8 of the time.

Statement 2: at some point HT will come up on consecutive throws. The next throw will be H or T with equal probability and so HTH and HTT will have an equal chance of coming up.

Statement 3: the average number of throws to see HT is 4, and as we have seen, the game will finish on the next throw regardless of what it is, so the average number of flips ending in either HTH or HTT is 5.

Statement 4: this seems like it should be true, but it isn’t. On average it will take me 10 throws to see HTH but my friend only 8 throws to see HTT. This is because if my friend reaches HT, she will either finish on the next throw, or have an H to start off a new attempt at HTT. If I’m on HT I will either finish on the next throw or be back at square one.

Interestingly, if we make the target sequences differ earlier, for instance HHH vs HTT, then even statements 2 and 3 will be false. HTT would win 3 times out of 5, and the average number of flips when HHH wins would be 5 2/5, whereas the average number of flips when HTT wins would be 5 11/15.