Shame I've never seen it shared online. I was actually hoping the submitted article was a proof of this, but you can't have everything in life.

It's not hard to set up a bet that gives you an arbitrarily high chance of gaining money, despite an expected value of less than 1.

More fundamentally, root-mean-square is the norm induced by the expectation inner product in the space of random variables. Norms generalize the geometric notion of length, so intuitively RMS is an appropriate measure of the "stochastic distance" from the origin of a random walk after a set number of steps. RMS can likewise be used as an analogue for geometric length for other purposes in a stochastic context, e.g., in calculating the similarity dimension of fractal stochastic processes like Brownian motion.