Probability of selecting 4 female committee out of 19 men and 11 women

To find the probability of selecting a committee of 4 females from a group consisting of 19 men and 11 women, we can use **combinatorics**.

### Step-by-Step Solution

1. **Total population**:
   - Men: 19
   - Women: 11
   - Total: 19 + 11 = 30 people

2. **Choosing 4 females** from 11 women:
   This is a combination problem, and the formula for combinations is:
   \[
   \binom{n}{r} = \frac{n!}{r!(n-r)!}
   \]
   where \( n \) is the total number of elements and \( r \) is the number of elements chosen.

   So, the number of ways to choose 4 women from 11 women is:
   \[
   \binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330
   \]

3. **Choosing the rest** (0 men from 19):
   Since we are only selecting women and no men, we need to calculate the number of ways to choose 0 men from 19 men:
   \[
   \binom{19}{0} = 1
   \]

4. **Total number of ways** to select any 4 people from 30 people:
   \[
   \binom{30}{4} = \frac{30 \times 29 \times 28 \times 27}{4 \times 3 \times 2 \times 1} = 27,405
   \]

5. **Probability**:
   The probability is the ratio of favorable outcomes (selecting 4 women) to the total outcomes (selecting any 4 people):
   \[
   P(\text{4 women}) = \frac{\binom{11}{4} \times \binom{19}{0}}{\binom{30}{4}} = \frac{330 \times 1}{27,405} = \frac{330}{27,405} \approx 0.01204
   \]

Thus, the probability of selecting a committee of 4 females is approximately **0.01204** or **1.204%**.

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Would you like more details or have any questions about this problem?

Here are some related questions for further exploration:
1. What is the probability of selecting 3 females and 1 male?
2. How does the probability change if there are 15 men and 15 women?
3. What is the probability of selecting at least 1 female?
4. How would the result change if we select a committee of 5 instead of 4?
5. How many different committees of 4 people can be formed from the group?

**Tip**: In probability problems involving selection, combinations are often used to find the total possible outcomes without worrying about the order.

Calculate the probability of selecting a committee of 4 females from a group of 19 men and 11 women. This problem uses combinatorics and probability formulas to determine a solution. Suitable for Grades 10-12, with a step-by-step breakdown and the key concept of combinations explained.

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