The problem with your analysis is that "go for the highest value" is not equivalent to "maximize expected value of the number you end up with" For example, con

The problem with your analysis is that "go for the highest value" is not equivalent to "maximize expected value of the number you end up with" For example, consider a similar game where you draw and then optionally re-draw (like in the original game) but the amount of money you win if you end up with a number p is p if p < 0.8 and p + b if p > 0.8, where b is some constant. Then you're also trying to "go for the highest value". However it is clear that in the limit of large b, your best strategy is to redraw if p < 0.8.