Question

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 event of selecting a man and the event of selecting a man the next time are dependent.

the probability of randomly selecting a four-person committee consisting entirely of men from a pool of 14 men and 15 women is
(round to four decimal places as needed.)

Explanation:

Classify the events

The selection of committee members is done without replacement. Selecting a man first changes the pool composition and thus alters the probability of selecting a man next. Therefore, the events are dependent.

Calculate total outcomes

Using Probability of Events, we find the total number of ways to choose a 4-person committee from the entire pool of 29 people (14 men + 15 women):

$$ \binom{29}{4} = \frac{29 \times 28 \times 27 \times 26}{4 \times 3 \times 2 \times 1} = 23,751 $$

Calculate favorable outcomes

We find the number of ways to choose 4 men from the 14 available men:

$$ \binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1,001 $$

Compute the probability

We divide the favorable outcomes by the total outcomes to find the probability:

$$ P = \frac{1,001}{23,751} \approx 0.0421456 $$

Round the result

Rounding \(0.0421456\) to four decimal places yields \(0.0421\).

Answer:

Question 1

The event of selecting a man and the event of selecting a man the next time are dependent.

Question 2

The probability of randomly selecting a four-person committee consisting entirely of men from a pool of 14 men and 15 women is 0.0421.