Pre Calculus word problem

Question

Suppose 14 men and 24 women apply for jobs. If four people are selected randomly to fill job positions, what is the probability only one man was selected for a job?

Answer 1

 

Hi Violet,

 

1) The probability of selecting a man first and then three women:

    (14/38)·(24/37)·(23/36)·(22/35) = 0.096

 

2) The probability of selecting a woman, a man, and then two women:

    (24/38)·(14/37)·(23/36)·(22/35) = 0.096

 

3) The probability of selecting two women, a man, and then a woman:

    (24/38)·(23/37)·(14/36)·(22/35) = 0.096

 

4) The probability of selecting three women and then a man:
    (24/38)·(23/37)·(22/36)·(14/35) = 0.096

 

So, the answer is 0.384 (= 0.096 + 0.096 + 0.096 + 0.096).

 

 

 

Answer 2

Suppose three women are chosen for the first three slots and a man for the final one:

 

²⁴/₃₈ is the probability of a woman being chosen for the first slot (24 possible women out of 38 total people)

²³/₃₇ is the probability of a woman being chosen for the second slot (there are now only 23 women left because one was already picked, and a total of 37 people left)

²²/₃₆ is similarly the probability for a 3rd woman being picked

¹⁴/₃₅ is the probability for a man being chosen for the final slot since there are 14 possible men and only 35 total people left.

 

We multiply the probabilities for the four events and end up with: ²⁴/₃₈ * ²³/₃₇ * ²²/₃₆ * ¹⁴/₃₅

Let's place the numerators together and denominators together for one big fraction. I'll explain why in a second:

We get: ^{(24*23*22*14)} / _{(38*37*36*35).}

 

This is our answer when the man is picked for the last slot. What if a man is picked for say the first slot and then three women for final three slots, would that change our result? The answer is no, it wouldn't because we end up with probabilities ¹⁴/₃₈ * ²⁴/₃₇ * ²³/₃₆ * ²²/_35. The numerators are in a different order but multiplication is commutative and so the result is the same as before.

 

(24*23*22*14) / (38*37*36*35) = .095969654

Probability is ~ 9.60%