A committee of three persons is to be randomly selected from a group of three men and two women and the chair person will be randomly selected from 2 woman and 1 men for the committee. The probability that the committee will have exactly two woman and one man and that the chair person will be a woman, is

Options

A. `(1)/(5)`
B. `(8)/(15)`
C. `(2)/(3)`
D. `(3)/(10)`

Explanation

To solve the problem, we need to find the probability that a committee of three persons selected from a group of three men and two women will consist of exactly two women and one man, and that the chairperson will be one of the women.

### Step-by-Step Solution:

1. **Identify the Total Number of People**:
We have 3 men and 2 women, making a total of 5 people.

2. **Determine the Event of Interest (E)**:
We want to find the probability of selecting exactly 2 women and 1 man for the committee.

3. **Calculate the Number of Ways to Select 2 Women and 1 Man**:
- The number of ways to choose 2 women from 2 is given by \( \binom{2}{2} = 1 \).
- The number of ways to choose 1 man from 3 is given by \( \binom{3}{1} = 3 \).
- Therefore, the total number of ways to select 2 women and 1 man is:
\[ N(E) = \binom{2}{2} \times \binom{3}{1} = 1 \times 3 = 3. \]

4. **Calculate the Total Number of Ways to Select 3 People from 5**:
- The total number of ways to select any 3 people from 5 is given by \( \binom{5}{3} = 10 \).

5. **Calculate the Probability of Event E**:
- The probability of selecting 2 women and 1 man is:
\[ P(E) = \frac{N(E)}{N(S)} = \frac{3}{10}. \]

6. **Determine the Event of Interest for Chairperson (F)**:
- We need to find the probability that the chairperson selected from the committee (which consists of 2 women and 1 man) is a woman.
- The number of ways to select a chairperson (woman) from the committee is 2 (since there are 2 women).
- The total number of committee members is 3 (2 women + 1 man).

7. **Calculate the Probability of Event F**:
- The probability that the chairperson is a woman is:
\[ P(F) = \frac{2}{3}. \]

8. **Combine the Probabilities**:
- Since we need both conditions to be satisfied (the committee having 2 women and 1 man, and the chairperson being a woman), we multiply the probabilities:
\[ P(E \text{ and } F) = P(E) \times P(F) = \frac{3}{10} \times \frac{2}{3} = \frac{2}{10} = \frac{1}{5}. \]

### Final Answer:
The probability that the committee will have exactly 2 women and 1 man, and that the chairperson will be a woman, is \( \frac{1}{5} \).