The offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 17 candidates. Nine 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) $\mathrm{P}$ (all offices filled by debate team members) $=0.053$
(Round to three decimal places as needed.)
(b) $P$ (no offices filled by debate team members) $=$
(Round to three decimal places as needed.)
Answer
Final Answer: (a) The probability that all of the offices are filled by members of the debate team is \(\boxed{0.053}\) (rounded to three decimal places). (b) The probability that none of the offices are filled by members of the debate team is \(\boxed{0.029}\) (rounded to three decimal places).
Step 1 ：First, we need to calculate the total number of ways to fill the 4 offices from the 17 candidates. This is a combination problem, and can be calculated as \(C(17, 4)\), which equals 2380.
Step 2 ：For part (a), we need to calculate the probability that all of the offices are filled by members of the debate team. This is a combination problem, where we are choosing 4 people out of 9 (the number of debate team members). The number of ways to do this is \(C(9, 4)\), which equals 126.
Step 3 ：The probability for part (a) is then the number of ways to choose 4 debate team members divided by the total number of ways to fill the offices, or \(\frac{126}{2380}\), which equals 0.053 when rounded to three decimal places.
Step 4 ：For part (b), we need to calculate the probability that none of the offices are filled by members of the debate team. This is also a combination problem, where we are choosing 4 people out of 8 (the number of non-debate team members). The number of ways to do this is \(C(8, 4)\), which equals 70.
Step 5 ：The probability for part (b) is then the number of ways to choose 4 non-debate team members divided by the total number of ways to fill the offices, or \(\frac{70}{2380}\), which equals 0.029 when rounded to three decimal places.
Step 6 ：Final Answer: (a) The probability that all of the offices are filled by members of the debate team is \(\boxed{0.053}\) (rounded to three decimal places). (b) The probability that none of the offices are filled by members of the debate team is \(\boxed{0.029}\) (rounded to three decimal places).
