Question

Membership selection. A town council has 13 ​members, 7 Democrats and 6 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?

Answer 1

a) Here order of selection is important, so we will use permutation.

Total number of ways to choose president and​ vice-president = 13P2 = 13! / (13 - 2)! = 156

Number of ways to choose president and​ vice-president from democrats = 7P2 = 7! / (7 - 2)! = 42

P(both democrats) = 42 / 156 = 0.2692

b) Here order of selection is not important, so we will use combination.

Total number of ways to select 3 members = 13C3 = 13! / (3! * (13 - 3)!) = 286

P(Republicans make up the​ majority) = P(2 republicans and 1 democrats) + P(3 republicans)

                                                            = (6C2 * 7C1 / 286) + (6C3 / 286)

                                                            = (15 * 7 / 286) + (20 / 286)

                                                            = 0.4371