A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules: (I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position; (II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position. For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$. Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).
A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules:\\
(I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position;\\
(II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position.\\
For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$.\\
Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).