Question

The offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 12 candidates. Five of the candidates are members of the debate team. (a) What is the probability that all of the offices are filled by members of the debate team? (b) What is the probability that none of the offices are filled by members of the debate team? (a) P(all offices filled by debate team members) = (Round to three decimal places as needed.)

Answer 1

12 candidates

5 of the candidates are on debate team

4 members in the offices to be chosen

a) P(all members of the office are from debate team) = ?

n(all members of the office are from debate team) = $\binom{5}{4}$ = 5

n(ways members of the office can be chosen) = $\binom{12}{4}$ =

12+11+ 10 +9 | 4*3*2*1

n(ways members of the office can be chosen) = 11*5*9 = 495

P(all members of the office are from debate team) = n(all members of the office are from debate team)/n(ways members of the office can be chosen)

P(all members of the office are from debate team) = 5/495 = 1/99 = 0.010

b) P(none of the members of the office are from debate team) = ?

n(none of the members of the office are from debate team) =
ܗ ܠܢ
=
I *7*&* 9*9* 2 * 8

n(none of the members of the office are from debate team) = 2 * 7 * 5 = 70

n(ways members of the office can be chosen) = $\binom{12}{4}$ =

12+11+ 10 +9 | 4*3*2*1

n(ways members of the office can be chosen) = 11*5*9 = 495

P(none of the members of the office are from debate team) = n(none of the members of the office are from debate team)/n(ways members of the office can be chosen)

P(none of the members of the office are from debate team) = 70/495 = 14/99 = 0.141