How to calculate the maximum deviation of a point from zero on a particular step?

250-300 tests are conducted with two (A, B) coins. The second B (ideal) coin is introduced due to the possibility of dishonesty of the first coin. The coincidence / mismatch of the symbols ("common game") is tracked. After the completion of 1 test cycle, 2, 3, etc. (between cycles 1000 tests of coin A) are carried out. Previous cycles are forgotten. I draw a graph: match (move point up from axis 0), mismatch (move point down from axis 0). We obtain a random walk.

How to calculate the maximum deviation of a point from zero on a particular step?

 earlier I determined the standard deviation (1-sigma, 2-sigma, 3-sigma) √npq.  then I learned about the mathematical expectation of the modulus of the difference (+1; -1) | X-Y | .√2n / π (approximate formula).   what's right? can have other options?