Question for Mathematicians

Imagine a player with 20 units bankroll. He decides to bet one unit at advantage situations (average +1%).We know that standard deviation is 1.1 units per hand. For 100 bets this would be 10*1.1 = 11 units.So it
would take 2 SD to knock him out within the first hundred hands. The probalitity for 2 SD is a mere 2.3%. Now,does this mean that his risk of tapping out in that period is this 2.3%?
I don't think so because all the cases of being down by 20 or more units and coming back before the 100th trial are not included in this figure!
But what is the correct percentage?

Francis Salmon