Lets assume the two of a kind is of the number a. Below are the number of ways to advance closer and the number of corresponding combinations. So there are 15 ways to progress to a four of a kind. There is obviously only 1 way to progress to a five of a kind. Next solve for E 1 .
Lets assume that the one of a kind is already an a.
There is obviously 1 way to get to a yahtzee. So now we have four equations and four unknowns and can solve with matrix algebra. Here is the matrix that needs to be solved. Here are the final answers.