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 women from a pool of 10 men and 16 women

To determine if these events are independent or dependent, we first analyze the situation:

We are forming a four-person committee entirely of women from a pool containing 10 men and 16 women.

1. **Independence/Dependence of Events**:  
   Selecting a woman for the committee each time affects the subsequent selections, as it changes the pool size and the number of women remaining. Thus, each selection is dependent on the previous one, making these events **dependent**.

2. **Probability of the Combined Event**:  
   We want the probability that all four members selected are women. We calculate this probability step-by-step as each selection occurs:

   - **Total Pool**: 26 individuals (10 men + 16 women)
   - **First Selection (woman)**: Probability = \( \frac{16}{26} \)
   - **Second Selection (woman)**: Probability = \( \frac{15}{25} \) (since one woman has already been chosen)
   - **Third Selection (woman)**: Probability = \( \frac{14}{24} \)
   - **Fourth Selection (woman)**: Probability = \( \frac{13}{23} \)

   Now, we multiply these probabilities to find the probability of selecting four women in a row:

   \[
   P(\text{4 women}) = \frac{16}{26} \times \frac{15}{25} \times \frac{14}{24} \times \frac{13}{23}
   \]

   Let's calculate this:

   \[
   = \frac{16 \times 15 \times 14 \times 13}{26 \times 25 \times 24 \times 23} 
   = \frac{43680}{358800} 
   = \frac{121}{995} 
   \approx 0.1216
   \]

Therefore, the probability of selecting a four-person committee consisting entirely of women from this pool is approximately **0.1216**.

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Would you like more details on any part of the process, or have any further questions?

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### Related Questions:
1. How do we determine if events are independent or dependent?
2. How would the probability change if we included an equal number of men and women?
3. What would the probability be if we needed only three women instead of four?
4. How would we calculate the probability if we allowed men on the committee?
5. What if the selection were done with replacement? How would that affect probability?

### Tip:
In probability, events are **independent** if the outcome of one does not affect the other. If each selection changes the total pool, the events are typically **dependent**.

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 women from a pool of 10 men and 16 women.

Learn how to determine if events are independent or dependent when forming a committee and calculate the probability of selecting four women from a pool of 10 men and 16 women. This problem involves probability concepts, dependent events, and combinatorics, making it suitable for Grades 10-12.

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