from bag 1, to get a white and then a black, or a black then a white,
the prob is (5/8 x 3/7) + (3/8 x 5/7) {NB NO replacement, ball is not put back in bag}
for bag 2, the same prob is: (4/9 x 5/8) + (5/9 x 4/8)
totalling this and assuming it's 50:50 that one bag will be chosen to the other,
the odds are 1/2( (5/8 x 3/7) + (3/8 x 5/7)) + 1/2 ( (4/9 x 5/8) + (5/9 x 4/8)) =
0.5456
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