Question

What is the probability of selecting 3 people at random, where the crowd has three times as many men as women, and finding the probability of selecting all men and 2 women and 1 man?

Answer 1

To solve the problem, let's denote the number of men and women in the crowd. Given that there are three times as many men as women, we can let the number of women be \( W \) and the number of men be \( 3W \).

### Total number of people:
The total number of people in the crowd is:
\[
N = W + 3W = 4W
\]

### 1. Probability of selecting all men:
To find the probability of selecting 3 men from this crowd, we use combinations.

The total number of ways to select 3 people from \( 4W \) people is:
\[
\binom{4W}{3}
\]

The number of ways to select 3 men from \( 3W \) men is:
\[
\binom{3W}{3}
\]

Thus, the probability \( P(\text{all men}) \) is given by:
\[
P(\text{all men}) = \frac{\binom{3W}{3}}{\binom{4W}{3}}
\]
Calculating these, we have:
\[
\binom{3W}{3} = \frac{3W(3W-1)(3W-2)}{6}
\]
\[
\binom{4W}{3} = \frac{4W(4W-1)(4W-2)}{6}
\]
So,
\[
P(\text{all men}) = \frac{3W(3W-1)(3W-2)}{4W(4W-1)(4W-2)}
\]

### 2. Probability of selecting 2 women and 1 man:
To find the probability of selecting 2 women and 1 man, we again use combinations:

The number of ways to select 2 women from \( W \) women is:
\[
\binom{W}{2}
\]
The number of ways to select 1 man from \( 3W \) men is:
\[
\binom{3W}{1}
\]

Thus, the probability \( P(2 \text{ women, } 1 \text{ man}) \) is given by:
\[
P(2 \text{ women, } 1 \text{ man}) = \frac{\binom{W}{2} \cdot \binom{3W}{1}}{\binom{4W}{3}}
\]

Calculating these, we have:
\[
\binom{W}{2} = \frac{W(W-1)}{2}
\]
\[
\binom{3W}{1} = 3W
\]
So, we plug these into the formula:
\[
P(2 \text{ women, } 1 \text{ man}) = \frac{\left(\frac{W(W-1)}{2}\right) \cdot 3W}{\binom{4W}{3}}
\]
This simplifies to:
\[
P(2 \text{ women, } 1 \text{ man}) = \frac{3W^2(W-1)/2}{\binom{4W}{3}}
\]

### Final answers:
Now we have the probabilities expressed as:
1. \( P(\text{all men}) = \frac{3W(3W-1)(3W-2)}{4W(4W-1)(4W-2)} \)
2. \( P(2 \text{ women, } 1 \text{ man}) = \frac{3W^2(W-1)/2}{\binom{4W}{3}} \)

In summary, this approach details how to calculate the specified probabilities based on the ratio of men to women in the crowd.