Estimating Probability for first appearance of HT before HH and HH before HT using Markov chains

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Consider the case where a coin is tossed repeatedly, I win when HH appears for the first time and you win when HT appears for the first time.

I know that one of us will win when the first Head appears so we have a 50-50 chance of winning.

But I recently studied about Markov chain and wanted to know whether it will be applicable here and if it is, I want to know how you can use it calculate the probability who has more probability to win.
I'm new to Markov chains and please help me solve this question