Question

A small musical club consists of 17 musicians playing string instruments, 13 musicians who play the percussion instruments, and 15 musicians playing brass instruments. An orchestra of 10 instrumentalists to be formed randomly (with all subsets of 10 people equally likely). Let A, B, and C be the events that there are musicians playing strings, percussions, and brass instruments, respectively, in the orchestra.

a) Find the probability that there are exactly 6 musicians who play the string instruments in the orchestra.

b) Find the probability that the orchestra doesn’t consist of the people who play the string instruments and people who play the brass instruments.

c) Find the probability that the orchestra has at least one representative from each of these three groups of musicians. Hint: Try to find P(Ac ∪ Bc ∪ C c ) first and then P(A ∩ B ∩ C).

Answer 1

a)

probability that there are exactly 6 musicians who play the string instruments in the orchestra

=P(6 from 17 of playing string instruments and 4 from remaining 28)

=(₁₇C₆)*(₂₈C₄)/(₄₅C₁₀) =12376*20475/3190187286 =0.0794

b)

probability that the orchestra doesn’t consist of the people who play the string instruments and people who play the brass instruments =P(all 10 from 13 of percussion instruments )

=(₁₃C₁₀)/(₄₅C₁₀)=286/3190187286 =0.000000090

c)

probability that the orchestra has at least one representative from each of these three groups of musicians

1- P(Ac ∪ Bc ∪ C c )

=1-(P(Ac)+P(Bc)+P(Cc)-P(Ac n Bc)-P(Ac n Cc)-P(Bc n Cc)+P(Ac n Bc n Cc))

=1-((₂₈C₁₀)/(₄₅C₁₀)+(₃₂C₁₀)/(₄₅C₁₀)+(₃₀C₁₀)/(₄₅C₁₀)-(₁₅C₁₀)/(₄₅C₁₀)-(₁₃C₁₀)/(₄₅C₁₀)-(₁₇C₁₀)/(₄₅C₁₀)+0)

=1-0.0337 =0.9663