[AaronBrown]

Your reasoning is correct, but you're putting too much faith in the calculation.

If you always replenish your bankroll to $x, then your risk of ruin is zero (unless you lose all $x before replenishment). The risk of ruin calculation says when you have $x, if you keep playing regardless of bankroll, there is y% chance that you will go broke. Every time you win or lose any money, your risk of ruin changes.

Most people use this as a theoretical result only. If your bankroll is large enough that your risk of ruin is, say, 1%, most people can play comfortably without worrying about bankroll. For most people, that means they can play their best poker and enjoy life. If the risk of ruin gets up to, say, 25%, you take some remedial action: move to lower stakes, play more conservatively or get some more money.

In that case, your risk of ruin never was 1%, because you never planned to play without adjustment all the way to ruin. But the number gave you the confidence to ignore bankroll for a while. Similarly, the 25% wasn’t real either, it was just an alarm signal to say, “start worrying about your bankroll.” You then adjusted your play to reduce the risk of ruin to an acceptable level again.

The reason people do this kind of thing is to avoid thinking about bankroll every day and in every poker decision. You establish a comfort zone for risk of ruin, then live and play as if the risk of ruin is zero. When you get outside your comfort zone, you fix things to get back in. Worrying about bankroll every day isn’t fun, and worrying about bankroll in a poker decision hurts your play