Estimating the time until we obtain five-in-a-row?
Just for fun, I coded this up in Mathematica. I started with a $10 \times 10$ matrix initialized to all zeros. I then considered two processes throwing darts in the cells: (1)  a cell is chosen uniformly and independently at random from the set of all cells, and (2) a cell is chosen uniformly and independently at random from the set of all empty cells. When a  dart hits a cell, the value of that cell is set to $1$. A five-in-a-row can appear on a row, column, the diagonal, or on the antidiagonal.

For both processes, I ran $N = 20000$ simulations. Rounded up to the nearest integer, for process (1) we need $65$ dart throws on average to obtain a five-in-a-row. For process (2), the same number was $47$.

This is not exactly what you were asking for, but maybe you can at least compare the numbers you get with mine.