Question 1180230: Q-3: [5+5+5+5 Marks] The classification of faculty at a college is indicated in the following table:
Male Female Total
Professor 12 3 15
Associate professor 15 9 24
Assistant professor 18 8 26
Instructor 20 15 35
Total 65 35 100
a) If a committee of three faculty members is selected at random what is the probability that:
i) All females.
ii) A professor and two associate professors.
b) If a faculty member is selected at random from this college, find the probability that:
i) The faculty is male, given that the faculty is instructor.
ii) The faculty is not professor given that the faculty is female.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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
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