HHT and HTH in tossing a coin A coin is flipped infinitely until you or I win. If at any point, the last three tosses in the sequence are $HHT$, I win. If at any point, the last three tosses in the sequence are $HTH$, you win. Which sequence is more likely?
Unfortunately, this configuration does not seem like the ones as "$HHT$ versus $THH$" (where clearly only $HHT$ wins iff the first two occuring $H$ are consecutive). Of course, here we can still assume that $TT$ does not occur (as after such a thing the game restarts), but it does not seem to help me enough.
Any help appreciated!

(1) If you started by $HH$ then the 1st person wins.
(2) If you started by $HT$ you have 2 equal probabilties that the game restarts (or it is a tie) if you got a $T$, and the 2nd wins if you got an $H$
(3) If you started by $TH$ then you have equal probabilities that the 1st person wins if you got an $H$, and if you got a $T$ then part (2) happens.
(4) If you started by $TT$ then the game restarts (or it is a tie).
Now you may have completed your tree diagram
