Let us assume that a school pitch $60\ m$ wide and $100\ m$ long is divided into squares with a side of $1\ m$. Then the children start from the middle of the left side of the playing field ($0\ m$ long and $30\ m$ wide) run towards the right, moving on diagonal squares randomly. Determine the probability that after $100$ steps a randomly selected child has not left the playing field provided that it finished in the middle of the right side (children can run freely from the field and come back to them).
I don't konw how to start this task. I try to count $P(\max S_k<31, S_{100}=0)-P(\max S_k>=31,S_{100}=0)$, but it's harder than I thought.