Probability= =

Let X denote the number of moves by A.
Let Y denote the number of moves by B.

Pr(X = 1) = Pr(X = 2) = Pr(X = 3) = Pr(X = 4) = Pr(X = 5) = Pr(X = 6) = 1/6

Pr(Y = 1) = Pr(Y = 2) = Pr(Y = 3) = 1/3

The required probability is
Pr(Y > X)
= Pr(X < Y)
= Pr( (X, Y) ∈ { (1, 2), (1, 3), (2, 3) } )
= Pr(X = 1 ∩ Y = 2) + Pr(X = 1 ∩ Y = 3) + Pr(X = 2 ∩ Y = 3)
= Pr(X = 1)Pr(Y = 2) + Pr(X = 1)Pr(Y = 3) + Pr(X = 2)Pr(Y = 3)
= (1/6)(1/3) + (1/6)(1/3) + (1/6)(1/3)
= (1/6)(1/3)(3)
= 1/6