A father has 3 children with at least one boy. The probability that he has 2 boys and 1 girl is `1//4` b. `1//3` c. `2//3` d. none of these

Options

A. `1//4`
B. `1//3`
C. `2//3`
D. None of these

Explanation

To solve the problem, we need to find the probability that a father has 2 boys and 1 girl given that he has at least 1 boy. Let's break this down step by step.

### Step 1: Define the Sample Space
We start by defining the sample space for the gender combinations of 3 children. The possible combinations of boys (B) and girls (G) are:
1. BBB (3 boys)
2. BBG (2 boys, 1 girl)
3. BGB (2 boys, 1 girl)
4. GBB (2 boys, 1 girl)
5. BGG (1 boy, 2 girls)
6. GBG (1 boy, 2 girls)
7. GGB (1 boy, 2 girls)
8. GGG (3 girls)

Since we know that there is at least one boy, we can eliminate the combinations that do not meet this criterion. Thus, the relevant sample space (A) becomes:
1. BBB
2. BBG
3. BGB
4. GBB
5. BGG
6. GBG
7. GGB

### Step 2: Identify Event B
Event B is the event that the father has 2 boys and 1 girl. In our sample space, the combinations that satisfy this condition are:
1. BBG
2. BGB
3. GBB

So, there are 3 outcomes that correspond to event B.

### Step 3: Count the Outcomes
Now, we need to count the total number of outcomes in the sample space A (which includes at least one boy):
- Total outcomes in A = 7 (as listed above)

### Step 4: Count the Outcomes for A ∩ B
Next, we count the outcomes that are in both A and B (i.e., having at least one boy and having 2 boys and 1 girl):
- Outcomes in A ∩ B = 3 (BBG, BGB, GBB)

### Step 5: Calculate the Probability
Now we can use the formula for conditional probability:
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]

Where:
- \( P(A \cap B) = \frac{\text{Number of outcomes in } A \cap B}{\text{Total outcomes in sample space}} = \frac{3}{8} \)
- \( P(A) = \frac{\text{Number of outcomes in } A}{\text{Total outcomes in sample space}} = \frac{7}{8} \)

Thus,
\[ P(B | A) = \frac{3/8}{7/8} = \frac{3}{7} \]

### Final Step: Conclusion
The probability that the father has 2 boys and 1 girl given that he has at least one boy is \( \frac{3}{7} \).

### Answer
Since \( \frac{3}{7} \) is not listed among the options, the answer is **d. none of these**.

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