Coin Series #3

You are bored and you keep on tossing a fair coin. What is the probability that you have a sequence of H HH (heads heads heads) before THH (tails heads heads)? Prove the outcome by using Markov chains. The solution without using a Markov Chain is easy to find (as already suggested in the discussion section of this problem). However, sometimes you can be asked to solve a problem using a particular method, or, it can simply be good to exercise a problem using different methods in order to polish your skills for each approach. For this question, try to solve it using Markov Chains.

Answer format: round to 3 decimal places.