Spoiler for

Let A be the event that at least two white pawns are selected, and let B be the event that at least one pawn is selected. We seek P(A|B). Since A implies B, we have P(A and B) = P(A). So P(A|B) = P(A and B)/P(B) = P(A)/P(B). So we just need to calculate P(A) and P(B), which is fairly straightforward.

P(A) = 1-P(less than two white pawns selected) = 1-P(exactly one white pawn selected)-P(no white pawns selected) = 1-24*23*22*21*40/32^{5}-24*23*22*21*20/32^{5} = 1-24*23*22*21*60/32^{5} = 1-23*11*7*5*3^{3}/2^{19} = 285203/524288

P(B) = 1-P(no pawns selected) = 1-16*15*14*13*12/32^{5} = 1-13*7*5*3^{2}/2^{18} = 258049/262144

P(A)/P(B) = 285203/516098, which would be the probability we seek.