Determine whether the following individual 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 14 men and 15 women
The probability of randomly selecting a four-person committee consisting entirely of men can be found by dividing the number of favorable outcomes (selecting 4 men) by the total number of possible outcomes (selecting any 4 people).
First, let's determine whether the events are independent or dependent.
If we were to randomly select the first person and that person turned out to be a man, the probability of the second person being a man would change since there would be one less man in the pool. Therefore, the events are dependent.
To find the probability of the combined event, we will use the concept of conditional probability.
P(1st man) = number of men / total number of people = 14 / (14 + 15) = 14/29
For the second person, we have:
P(2nd man | 1st man) = (number of men - 1) / (total number of people - 1) = (14 - 1) / (29 - 1) = 13/28
For the third person:
P(3rd man | 2nd man) = (number of men - 2) / (total number of people - 2) = (14 - 2) / (29 - 2) = 12/27
For the fourth person:
P(4th man | 3rd man) = (number of men - 3) / (total number of people - 3) = (14 - 3) / (29 - 3) = 11/26
Since the events are dependent, we need to multiply these probabilities together:
P(combined event) = P(1st man) * P(2nd man | 1st man) * P(3rd man | 2nd man) * P(4th man | 3rd man)
= (14/29) * (13/28) * (12/27) * (11/26)
≈ 0.052 (rounded to three decimal places)
Therefore, the probability of randomly selecting a four-person committee consisting entirely of men is approximately 0.052 or 5.2%.