To calculate the probability of selecting a four-person committee consisting entirely of women from a pool of 13 men and 19 women, follow these steps:

1. Total possible committees: The total number of ways to select 4 people from the pool of $13 + 19 = 32$ people is given by the combination formula $C(n, k)$, where $n$ is the total number of people, and $k$ is the number of people selected:

$C(32, 4) = \frac{32!}{4!(32 - 4)!} = \frac{32!}{4!28!}$

2. Ways to select 4 women: The number of ways to select 4 women from the 19 women is:

$C(19, 4) = \frac{19!}{4!(19 - 4)!} = \frac{19!}{4!15!}$

3. Probability: The probability is the ratio of favorable outcomes (selecting 4 women) to the total possible outcomes (selecting 4 people from 32):

$P = \frac{C(19, 4)}{C(32, 4)}$

Now let's compute these values.

Step 1: Calculate $C(32, 4)$ and $C(19, 4)$.

$C(32, 4) = \frac{32 \times 31 \times 30 \times 29}{4 \times 3 \times 2 \times 1} = 35960$

$C(19, 4) = \frac{19 \times 18 \times 17 \times 16}{4 \times 3 \times 2 \times 1} = 3876$

Step 2: Compute the probability.

$P = \frac{3876}{35960} \approx 0.1078$

Thus, the probability of selecting a four-person committee consisting entirely of women is approximately 0.1078.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the probability of selecting 4 men from the same pool?
2. How does the probability change if the committee size is 3 people?
3. What is the total number of different four-person committees possible from the group?
4. How would you calculate the probability if the selection was based on men instead of women?
5. How does the probability change if there are fewer women in the pool?

Tip: Always check if the event in question involves combinations or permutations, as it significantly changes the approach to solving the problem.