Theory question 107
ProbabilityA bag contains 12 white balls and 8 black balls, another contains 10 white balls and 15 black balls. If two balls are drawn, without replacement, from each bag, find the probability that (a) all four balls are black (b) exactly one of the four balls is white.
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(a) Bag 1 both black: 8/20 × 7/19 = 14/95; Bag 2 both black: 15/25 × 14/24 = 7/20
P(all black) = 14/95 × 7/20 = 49/950
(b) P(exactly one white from bag 1) = 2 × 12/20 × 8/19 = 48/95; P(exactly one white from bag 2) = 2 × 10/25 × 15/24 = 1/2
P(exactly one white) = P(1 white from bag 1, both black from bag 2) + P(both black from bag 1, 1 white from bag 2)
= 48/95 × 7/20 + 14/95 × 1/2 = 336/1900 + 140/1900 = 476/1900 = 119/475 (≈ 0.251)
Treat each bag separately, then multiply; for 'exactly one' add the two ways it can happen.