Question

Given 10 democrats, 7 republicans, and 3 independents, what is the probability that AT LEAST ONE of the groups is not represented in a randomly selected committee of 5 people?

Answer 1

let A, B and C are probability that no democrat, no republican , no independent are in committee

P(A) =P(select all 5 from remaining 10 people =(₁₀C₅)/(₂₀C₅)

P(B) =P(select all 5 from remaining 13 people =(₁₃C₅)/(₂₀C₅)

P(C) =(₁₇C₅)/(₂₀C₅)

P(A n B) =(₃C₅)/(₂₀C₅) =0

P(A n C) =(₇C₅)/(₂₀C₅)

P(B n C )=(₁₀C₅)/(₂₀C₅)

P(A n B n C) =0

therefore probability that AT LEAST ONE of the groups is not represented in a randomly selected committee of 5 people =P(A u B u C) =P(A)+P(B)+P(C) -P(A n B)-P(A n C)-P(B nC)+P(A n B n C)

=(₁₀C₅)/(₂₀C₅)+(₁₃C₅)/(₂₀C₅)+(₁₇C₅)/(₂₀C₅)-0-(₇C₅)/(₂₀C₅) -(₁₀C₅)/(₂₀C₅)+0

=(252/15504)+(1287/15504)+(6188/15504)-(21/15504)-(252/15504)=7454/15504 =0.4808