# A congressional committee contains 3 democrats and 6 republicans.  A subcommittee of 3 is to be randomly seleted from all subcommittees of 3 which contain at least 1 democrat.  What is the...

A congressional committee contains 3 democrats and 6 republicans.  A subcommittee of 3 is to be randomly seleted from all subcommittees of 3 which contain at least 1 democrat.  What is the probability that this new subcommittee will contain at least 2 democrats?

llltkl | Student

Possible number of all 3 member subcommittees: 9C3 = (9*8*7*6) / (3*2*1) = 84
Number of 3 member subcommittees with 0 D and 3 R = (3C0) x (6C3) = 1*20 = 20
Number of 3 member subcommittees with 1 D and 2 R = (3C1) x (6C2) = 3 x 15 = 45

Now,

Number of subcommittees with at least 1 democrat
= Total number of all subcommittees – Number of subcommittees with 0D, 3R
= 84 – 20

= 64

Number of subcommittees with at least 2 democrats
= Total number of all subcommittees – Number of subcommittees with 0D, 3R – Number of subcommittees with 1D, 2R

= 84 - 20 - 45
= 19

Therefore, Probability of choosing a subcommittee with at least 2 democrats from the subcommittees with at least 1 democrat
= 19/64
= 0.297