Christmas Contest - Individual Round - Problem 5

Jerrily was playing a game of tetr.io against Alvin and Richard. Normally, each player has a \dfrac{1}{3} chance of winning.

Sometimes Jerrily gets very triggered when Alvin Pan says “grr” to him. This causes him to become egotistical and attempt stick spin openers and t-spins.

By doing this, Jerrily decreases his probability of winning by 50\% (that is, his probability is multiplied by \frac{1}{2}). Alvin has a \dfrac{3}{4} chance to aggravate Jerrily.

If the expected chance of Jerrily winning is denoted by \dfrac{m}{n}, where m and n are positive co-prime integers, compute m + n.