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(A) P=0.22 or 22% (B) P=50%
(A) We have a tree of probabilities, with two steps: president and vice-president.
If the president is elected first, there is a 5/10 chances it will be Democrat.
If this happened, in the election of vice-president, there is 4/9 chances it will be a Democrat chosen again, because there is one Democrat member already selected for President.
So the chances of having a Democrat president and vice president are:
(5/10) x (4/9) = 20/90 = 0.22 = 22%.
(B) We have another tree, with tree steps. The Republicans have majority whenever there are 2 or 3 members in the 3-person comitee.
It can be calculated as the sum of the probability of having 3 republican members (RRR) and the probability of having 2 republican members (RRD, RDR and DRR).
P(RRR) = (5/10) x (4/9) x (3/8) = 1/9
P(RRD) = (5/10) x (4/9) x (5/8) = 1/7
P(RDR) = (5/10) x (5/9) x (4/8) = 1/7
P(DRR) = (5/10) x (5/9) x (4/8) = 1/7
P(RRR) + P(RRD) + P(RDR) + P(DRR) = 1/9 + 1/7 + 1/7 + 1/7 = 1/2 = 50%
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