Well it is interesting, I assume that the phrasing is important "the first black ball". So if you say the 2nd draw, and they've already drawn one on the first one then you lose.
Your probability of drawing a black ball in the first draw is 2/10 = .20
The probability of drawing your first black ball on the second draw is the probability of drawing a white ball first and then a black ball = (8/10)(2/9) = 0.178
Probability of drawing your first black ball on the third turn = (8/10)(7/9)*(2/8) = 0.156