Membership selection. A town council has 10 members, 5 Democrats and 5 Republicans. (A) If the president and vice-president are selected at random, what is the probability that they are both Democrats? (B) If a 3-person committee is selected at random, what is the probability that Republicans make up the majority?
Accepted Solution
A:
Answer:(A) P=0.22 or 22% (B) P=50% Step-by-step explanation:(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) = 1/9P(RRD) = (5/10) x (4/9) x (5/8) = 1/7P(RDR) = (5/10) x (5/9) x (4/8) = 1/7P(DRR) = (5/10) x (5/9) x (4/8) = 1/7P(RRR) + P(RRD) + P(RDR) + P(DRR) = 1/9 + 1/7 + 1/7 + 1/7 = 1/2 = 50%