I'm thinking that's wrong and that you should make a probability 
tree diagram of all possible outcomes, with the probability of 
going along each branch (line) written on the branch:



The successful outcomes are boxed in red.

The probability of getting the highest highlighted 4 on the 
right is to go along the branch from START to 1 (roll 1 first),
with probability 1/4, and then go along the branch from 1 to 
the 4 (roll 4 second), with another probability of 1/4.  So
the probability of getting the 4 by rolling a 1 first and then
a 4, is gotten by multiplying the 1/4 to go from START to 1, times
the 1/4 to go from the 1 to the 4. It is similar for the other 
outcomes on the far right

So 

the probability of the upper highlighted 4 is (1/4)(1/4) = 1/16.
the probability of the upper highlighted 5 is (1/4)(1/4) = 1/16.
the probability of the other highlighted 4 is (1/4)(1/4) = 1/16.
the probability of the other highlighted 5 is (1/4)(1/4) = 1/16.
the probability of the highlighted 6 is (1/4)(1/4) = 1/16.

plus

the probability of the highlighted 4 on the bottom, the case where
a 4 was rolled first and then you stopped is 1/4, so the desired
probability is the sum of all those probabilities:

1/16 + 1/16 + 1/16 + 1/6 + 1/16 + 1/4 = 9/16.

Edwin