Probability that we see the pattern HHT before HTH in tossing a coin

Let p is the probability HHT comes before HTH. (STATE ZERO)

p = 1*(0.5)² (HH -> WINNING SCENARIO) + 0*(0.5)³ (HTH -> LOSING SCENARIO) + (0.5)³p (HTT -> GAME RESETS) + 0.5P (T -> GAME RESETS)

AFTER SOLVING, p = 2/3

A simple way of looking at it is this way, Once we flip an H,

Possible outcomes for next two tosses are HT -> WINNING SCENARIO

HH -> WINNING SCENARIO

TH -> LOSING SCENARIO

TT -> GAME RESETS

Therefore, 2 out of 3 outcomes(which give a result) are winning scenarios. Hence, p -> 2/3