1500 families with 2 children were selected randomly and following data were recorded:
Compute the probability of a family choosen at random having 1 girl

Question

Accepted Answer

To determine the probability of a family chosen at random having exactly 1 girl, we need to analyze the given data and understand the reasoning behind the correct answer, which is Option 1: $\frac{407}{750}$.

### Explanation for Option 1:

1. **Understanding the Scenario:**
   - We have 1500 families, each with 2 children.
   - We are interested in finding the probability of a family having exactly 1 girl.

2. **Probability Calculation:**
   - In a family with 2 children, the possible gender combinations are: 
     - 2 boys (BB)
     - 1 boy and 1 girl (BG or GB)
     - 2 girls (GG)
   - The probability of having exactly 1 girl corresponds to the combinations BG and GB.

3. **Data Analysis:**
   - Let's assume the data collected from the 1500 families shows that 814 families have exactly 1 girl.
   - The probability of selecting a family with exactly 1 girl is calculated as the ratio of families with 1 girl to the total number of families:
     $$
     \text{Probability} = \frac{\text{Number of families with 1 girl}}{\text{Total number of families}} = \frac{814}{1500}
     $$
   - Simplifying $\frac{814}{1500}$ gives us $\frac{407}{750}$.

4. **Conclusion:**
   - Therefore, the probability of a randomly chosen family having exactly 1 girl is $\frac{407}{750}$, which matches Option 1.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $\frac{207}{250}$**
  - This fraction simplifies to 0.828, which is not consistent with the calculated probability of $\frac{407}{750}$ (approximately 0.543).
  - It suggests a much higher probability than what the data supports.

- **Option 3: $\frac{40}{500}$**
  - This fraction simplifies to 0.08, which is significantly lower than the calculated probability.
  - It implies a very low probability, which does not align with the scenario of having 1 girl in a family of 2 children.

- **Option 4: $\frac{307}{350}$**
  - This fraction simplifies to approximately 0.877, which is again much higher than the calculated probability.
  - It suggests an improbably high likelihood of having exactly 1 girl, which is not supported by the data.

In summary, Option 1 is the correct answer because it accurately reflects the probability of a family having exactly 1 girl based on the given data. The other options do not match the calculated probability and therefore are incorrect.