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

The political committee consists of M=8 Democrats, N=8+5=13 members. A subcommittee of size K=9 needs to be selected from N=13

Total number ways to select 9 members from 13 is

Let X be the number of democrats in the subcommittee. The number of ways to select X=x democrats from M=8 democrats is

If we select X=x democrats, the rest 9-x must be republicans. The number of ways to select 9-x republicans from 5 republicans is

The probability that this subcommittee will consist of X=x Democrats is

X can also be considered to have a Hypergeometric distribution with parameters M=8,N=13,K=9.

The probability that this subcommittee will consist of X=x Democrats is given by the hypergeometric distribution,

a) The probability that this subcommittee will consist of X=5 Democrats if they were randomly selected is

ans: The probability that this subcommittee will consist of five Democrats and four Republicans if they were randomly selected is 0.3916

b) To calculate the mean, we can either use the formula for the expected value of X for a Hyper Geometric distribution or calculate using the pmf of X

Method 1: Using the formula

Method 2: If we have to use the pmf, first we will find the P(x) for x=4,5,..,8

x	P(x)
4	0.0979
5	0.3916
6	0.3916
7	0.1119
8	0.0070

The expected value of X is

ans: The mean of this distribution is 5.538

c)

Method 1: The standard deviation using the formula for Hyper Geometric distribution is

Method 2: Alternatively, using the pmf, first we get the expectation of

The standard deviation of X is

ans: The standard deviation of this distribution is 0.843