The possible state of the game is that the player has 0, 1, 2, ..., 6 dollars.

The 0 and 6 dollar states are absorbing.

So lets represent the game after plays as a column vector the th element being the probability that the game is in sate . Then after the th play the probabilities satisfy::

which give the 1-step transitions form which you should be able to calculate the 1-step transition matrix.

CB