[jcl]

Yes, it’s a mistake to take the expected value before the ball is drawn, because the drawn ball adds information, disproportionately changing the expected value.

Consider an alternate game, where I flip a hidden coin… if it is heads, I let you pick from an urn with 99 red and 1 green; if it is tails, I let you pick from an urn with 1 red and 99 green. The expected number of red is still 50, and the chance of drawing that first red is still 50%. But once that red is drawn, your expectation of what the next draw will be should change significantly.