I think you're working from the view that you're starting from the middle of the game. But you're not, you always start at the beginning. The probability of every possible selection is weighted by the probability of the events that bring you to that selection.
You can say 'if the 5 blacks are chosen in the first 20 selections then you have 100% chance of getting a white' but that's conditional on that precondition happening, which itself has a less than 100% probability of happening.
Why not try this thought experiment. Imagine there are 2 marbles, 1 black and 1 white. You can go first or second. Which do you choose? Is it better to later? No.
