Question about a simple game

[Thomas Okken]

That sounds like a one-dimensional random walk, where the expected distance from the starting point is indeed proportional to the square root of the number of steps.

I remember this being covered in calculus of probabilities in college, but I don't recall if we were expected to be able to derive the proof on our own, and I certainly don't remember the proof itself after all these years. But you're definitely on the right track!