determine whether the following events are independent or dependent. Then find the probability of the combined event.
Randomly selecting a four-person committee consisting entirely of men from a pool of 17 men and 12 women
The events in this scenario are dependent.
To find the probability of randomly selecting a four-person committee consisting entirely of men, we need to calculate the probability of selecting four men from the pool of 17 men.
P(selecting 4 men) = (Number of favorable outcomes) / (Number of possible outcomes)
Number of favorable outcomes:
Since we want to select a four-person committee consisting entirely of men, we need to choose 4 men from the pool of 17 men. This can be calculated using combination formula:
C(17, 4) = 17! / (4!(17-4)!)
= 17! / (4!13!)
= (17*16*15*14) / (4*3*2*1)
= 2380
Number of possible outcomes:
To find the number of possible outcomes, we need to choose 4 people from the total pool of men and women (17 men + 12 women).
Total possible outcomes = (17 + 12)C4
= 29C4
= 29! / (4!(29-4)!)
= 29! / (4!25!)
= (29*28*27*26) / (4*3*2*1)
= 20,475
Probability of selecting a four-person committee consisting entirely of men:
P(selecting 4 men) = Number of favorable outcomes / Number of possible outcomes
= 2380 / 20,475
≈ 0.1165 or 11.65%