Three groups of workers contain 3 men and one woman, 2 men and 2 women and 1 man and 3 women respectively. One worker is selected at random from each group. What is the probability that the group selected consists of 1 man and 2 women?

Answer

To solve the problem step by step, we need to calculate the probability of selecting a group consisting of 1 man and 2 women when one worker is selected from each of the three groups.

### Step-by-Step Solution:

1. **Identify the Groups and Their Composition:**
- Group 1: 3 men, 1 woman
- Group 2: 2 men, 2 women
- Group 3: 1 man, 3 women

2. **Determine the Total Number of Workers in Each Group:**
- Group 1: 4 workers (3 men + 1 woman)
- Group 2: 4 workers (2 men + 2 women)
- Group 3: 4 workers (1 man + 3 women)

3. **Identify Possible Cases for Selecting 1 Man and 2 Women:**
- Case 1: Man from Group 1, Woman from Group 2, Woman from Group 3
- Case 2: Woman from Group 1, Man from Group 2, Woman from Group 3
- Case 3: Woman from Group 1, Woman from Group 2, Man from Group 3

4. **Calculate the Probability for Each Case:**

**Case 1:**
- Probability of selecting 1 man from Group 1:
\[ P(\text{Man from G1}) = \frac{3}{4} \]
- Probability of selecting 1 woman from Group 2:
\[ P(\text{Woman from G2}) = \frac{2}{4} = \frac{1}{2} \]
- Probability of selecting 1 woman from Group 3:
\[ P(\text{Woman from G3}) = \frac{3}{4} \]
- Combined probability for Case 1:
\[ P(\text{Case 1}) = \frac{3}{4} \times \frac{1}{2} \times \frac{3}{4} = \frac{9}{32} \]

**Case 2:**
- Probability of selecting 1 woman from Group 1:
\[ P(\text{Woman from G1}) = \frac{1}{4} \]
- Probability of selecting 1 man from Group 2:
\[ P(\text{Man from G2}) = \frac{2}{4} = \frac{1}{2} \]
- Probability of selecting 1 woman from Group 3:
\[ P(\text{Woman from G3}) = \frac{3}{4} \]
- Combined probability for Case 2:
\[ P(\text{Case 2}) = \frac{1}{4} \times \frac{1}{2} \times \frac{3}{4} = \frac{3}{32} \]

**Case 3:**
- Probability of selecting 1 woman from Group 1:
\[ P(\text{Woman from G1}) = \frac{1}{4} \]
- Probability of selecting 1 woman from Group 2:
\[ P(\text{Woman from G2}) = \frac{2}{4} = \frac{1}{2} \]
- Probability of selecting 1 man from Group 3:
\[ P(\text{Man from G3}) = \frac{1}{4} \]
- Combined probability for Case 3:
\[ P(\text{Case 3}) = \frac{1}{4} \times \frac{1}{2} \times \frac{1}{4} = \frac{1}{32} \]

5. **Total Probability of Selecting 1 Man and 2 Women:**
- Combine the probabilities from all cases:
\[ P(\text{1 Man and 2 Women}) = P(\text{Case 1}) + P(\text{Case 2}) + P(\text{Case 3}) = \frac{9}{32} + \frac{3}{32} + \frac{1}{32} = \frac{13}{32} \]

### Final Answer:
The probability that the group selected consists of 1 man and 2 women is \( \frac{13}{32} \).