Question #41034

A system has two states, $S_1$ and $S_2$. It can undergo two types of transition protocols, $X$ and $Y$. Under protocol $X$, a system in $S_1$ transitions to $S_2$ with probability $\frac{1}{2}$ and remains in $S_1$ with probability $\frac{1}{2}$. A system in $S_2$ remains in $S_2$ with probability $1$. Under protocol $Y$, a system in $S_1$ remains in $S_1$ with probability $1$. A system in $S_2$ transitions to $S_1$ with probability $\frac{1}{2}$ and remains in $S_2$ with probability $\frac{1}{2}$. A fair coin is tossed. If it shows Heads, the system undergoes the sequence of protocols $X, Y, X, Y$ in that order. If it shows Tails, the system undergoes the sequence $X, X, Y, Y$ in that order. Given that the system initially starts in state $S_1$ and is found to be in state $S_1$ after the four protocols are completed, what is the probability that the coin showed Heads?