Question 1180230
**a) Committee of Three Faculty Members:**

**i) Probability of all females:**

* Total number of females: 35
* Total number of faculty: 100
* Number of ways to choose 3 females: ³⁵C₃ = (35 \* 34 \* 33) / (3 \* 2 \* 1) = 6545
* Number of ways to choose any 3 faculty: ¹⁰⁰C₃ = (100 \* 99 \* 98) / (3 \* 2 \* 1) = 161700
* Probability: 6545 / 161700 ≈ 0.0405

**ii) Probability of a professor and two associate professors:**

* Number of ways to choose 1 professor: ¹⁵C₁ = 15
* Number of ways to choose 2 associate professors: ²⁴C₂ = (24 \* 23) / (2 \* 1) = 276
* Number of ways to choose 1 professor and 2 associate professors: 15 \* 276 = 4140
* Probability: 4140 / 161700 ≈ 0.0256

**b) Single Faculty Member Selected:**

**i) Probability of male, given instructor:**

* Number of male instructors: 20
* Total number of instructors: 35
* Probability: 20 / 35 ≈ 0.5714

**ii) Probability of not professor, given female:**

* Total number of female faculty: 35
* Number of female professors: 3
* Number of female faculty who are not professors: 35 - 3 = 32
* Probability: 32 / 35 ≈ 0.9143