Two players, $A$ and $B$, play a sequential game with a fair six-sided die. They take turns rolling the die, starting with $A$. The game has at most two rounds (each player rolling at most twice).

• In Round 1, $A$ wins if they roll a 5 or 6. If $A$ does not win, $B$ rolls and wins if they roll a 4, 5, or 6.
• If $B$ does not win, Round 2 begins. $A$ rolls and wins if they roll any number $\ge 3$. If $A$ does not win, $B$ rolls and wins if they roll any number $\ge 2$. If no one wins after these rolls, the game is a draw. What is the probability that $B$ wins the game?