First compute the probabilities for a chip dropped in slot A using a tree diagram.

So

P(A' | A) = 1/4 + 1/8 = 3/8
P(B' | A) = 1/4 + 1/8 + 1/8 = 1/2
P(C' | A) = 1/8

To use a Pascal's Triangle approach for counting paths, we use 5 rows in Pascal's Triangle.  Again, the yellow portion of the table is actually on the game board.  Note that the first movement of the chip is forced to the right.

 A B C 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 A' B' C'

Looking at the 2nd row of Pascal's Triangle, we see that the 1 at the left end is outside the board.  Thus, we can assume that the chip that would have come from that cell  will not get to the 3rd row (or 4th or 5th rows), so to obtain the correct path count we subtract 1 from the 2 in the 3rd row of Pascal's Triangle and 1 from the 2nd 3 in the 4th row and 1 from the 2nd 4 in the 5th row.  Similarly, the a chip that would have come from the cells corresponding to 1 outside the board in the 3rd row and the 3 outside the board in the 4th row will not reach the 5th row, so we subtract 4 from the 6 in the 5th row.  So the approach gives the following table of path counts.

 A B C 1 1 1 1 2 -1 3 1 1 3 3 -1 2 1 1 4 6 -1 -3 2 4 -1 3 1 A' B' C'

Thus, our path count board looks like the following:

 A B C 1 1 1 1 2 1 2 3 1 A' B' C'

This agrees with the actual path count shown below.

Now we revisit the Pascal's Triangle approach for determining the probabilities.  As before, we reflect the 1 and 4  as shown.

Adding the bottom row, we obtain the denominator of the probability fractions which is 16 in this case.  The conditional probabilities are

P(A' | A) = 6/16 = 3/8
P(B' | A) = 8/16 = 1/2
P(C' | A) = 2/16 = 1/8

as we found before.

Note that for this case we do not obtain the correct values for the conditional probabilities by computing the ratios of the paths from A to A', B', and C', respectively, to the total number of paths.  This is expected because the paths are not equally likely.

Symmetry arguments give us the conditional probabilities

P(A' | C) = 1/8
P(B' | C) = 1/2
P(C' | C) = 3/8.