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 

menmen 

from a pool of 

1616 

men and 

1717 

women

Question content area bottom

Part 1

The event of selecting a 

manman 

and the event of selecting a 

manman 

the next time are 

dependent.

Part 2

The probability of randomly selecting a​ four-person committee consisting entirely of 

menmen 

from a pool of 

1616 

men and 

1717 

women is 

enter your response here.

​(Round to four decimal places as​ needed.)

### Part 1: Determine Whether the Events Are Dependent or Independent
Since we are selecting individuals from a pool without replacement, the probability changes as each individual is chosen. This means each selection depends on the previous one. Thus, the events are **dependent**.

### Part 2: Calculate the Probability of Selecting Four Men
We need to calculate the probability of choosing four men consecutively from a pool of 16 men and 17 women, making a total of 33 people. 

The probability of selecting a four-person committee of all men can be calculated as follows:

1. **Probability of selecting the first man:**  
   \[
   \frac{16}{33}
   \]

2. **Probability of selecting the second man:**  
   After selecting one man, there are now 15 men left out of 32 people remaining:
   \[
   \frac{15}{32}
   \]

3. **Probability of selecting the third man:**  
   After selecting two men, there are now 14 men left out of 31 people remaining:
   \[
   \frac{14}{31}
   \]

4. **Probability of selecting the fourth man:**  
   After selecting three men, there are now 13 men left out of 30 people remaining:
   \[
   \frac{13}{30}
   \]

5. **Combined probability of selecting four men consecutively:**  
   Multiply these probabilities together:
   \[
   \frac{16}{33} \times \frac{15}{32} \times \frac{14}{31} \times \frac{13}{30} = \frac{43680}{981240} \approx 0.0445
   \]

Therefore, the probability of selecting a four-person committee consisting entirely of men is approximately **0.0445** (rounded to four decimal places).

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Would you like a breakdown of each step or further explanation?

---

1. What if we chose four women instead of men?
2. How would the probability change if the pool had an equal number of men and women?
3. How does using replacement affect independence?
4. What if we wanted a committee of two men and two women?
5. How would this problem change if we increased the committee size to six?

**Tip:** Remember, events are dependent when each choice impacts the next!

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 16 men and 17 women.

Explore how to determine dependency in events and calculate the probability of forming a four-person committee of men from a pool of 16 men and 17 women. This solution includes detailed steps and calculations, focusing on probability concepts and the multiplication rule for dependent events. Suitable for high school students in Grades 10-12.

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