Go to the Right, Counter!

Discrete Mathematics Level 5

Suppose I play a game involving a counter and a \(5\)-square grid, as shown above.
Assign the values \(1\) to \(5\) to the squares in the grid from left to right.
I start with the counter on square \(1.\)

On each turn, I roll a standard six-sided die.
If the number that is rolled is less than or equal to the number (i.e. between \(1\) and \(5\) inclusive) of the square which the counter is on, I move the counter to the left one square. Otherwise, I move the counter to the right one square.

If I move the counter onto the green square, which is shown in the picture, I win.
However, if I move the counter onto the red square, I lose.

If the probability that I win the game is represented by \(\frac{A}{B},\) for positive coprime integers \(A\) and \(B,\) what is the value of \(B - A?\)


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