Question

A political committee consists of eight Democrats and five Republicans. A subcommittee of nine people needs to be formed from this group.​ (For this​ problem, define a success as a Democrat being selected for the​ subcommittee.) a. Determine the probability that this subcommittee will consist of five Democrats and four Republicans if they were randomly selected. b. Calculate the mean and standard deviation of this distribution. a. The probability that this subcommittee will consist of five Democrats and four Republicans if they were randomly selected is nothing. ​(Round to four decimal places as​ needed.) b. The mean of this distribution is nothing. ​(Round to three decimal places as​ needed.) The standard deviation of this distribution is nothing. ​(Round to three decimal places as​ needed.)

Answer 1

a.

Total number of people = 8 + 5 = 13

Number of ways to form a subcommittee of 9 people = 13C9 = 13! / (9! * (13 - 9)!) = 715

Number of ways to choose five Democrats from 8 = 8C5 = 56

Number of ways to choose four Republicans from 5 = 5C4 = 5

Probability that this subcommittee will consist of five Democrats and four Republicans = (8C5 * 5C4) / 13C9 = (56 * 5) / 715

= 0.3916

b.

Let X be the number of Democrats in the subcommittee of 9 People. Then possible values of X are 4, 5, 6, 7, 8

P(X = 4) = (8C4 * 5C5) / 13C9 = (70 * 1) / 715 = 0.0979

P(X = 5) = (8C5 * 5C4) / 13C9 = (56 * 5) / 715 = 0.3916

P(X = 6) = (8C6 * 5C3) / 13C9 = (28 * 10) / 715 = 0.3916

P(X = 7) = (8C7 * 5C2) / 13C9 = (8 * 10) / 715 = 0.1119

P(X = 8) = (8C8 * 5C1) / 13C9 = (1 * 5) / 715 = 0.0070

Mean of this distribution = 4 * 0.0979 + 5 * 0.3916 + 6 * 0.3916 + 7 * 0.1119 + 8 * 0.0070

= 5.538

E(X²) = 4² * 0.0979 + 5² * 0.3916 + 6² * 0.3916 + 7² * 0.1119 + 8² * 0.0070 = 31.3851

Var(X) = E(X²) - E(X)² = 31.3851 - 5.538² = 0.715656

Standard deviation of this distribution = = 0.846