1500 families with 2 children were selected randomly , and the following data were recorded.
`{:("Number of girls in a family",2,1,0),("Number of families",475,814,211):}`
Compute the probability of a family , chosen at random , having
No girls. Also check whether the sum of these probabilities is 1.

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the total number of families
The total number of families surveyed is given as 1500.

### Step 2: Identify the number of families with no girls
From the data provided, the number of families with no girls (0 girls) is 211.

### Step 3: Calculate the probability of a family having no girls
The probability \( P \) of a family having no girls can be calculated using the formula:
\[ P(\text{no girls}) = \frac{\text{Number of families with no girls}}{\text{Total number of families}} = \frac{211}{1500} \]

### Step 4: Calculate the probability of a family having one girl
The number of families with one girl is given as 814. Therefore, the probability of a family having one girl is:
\[ P(\text{one girl}) = \frac{814}{1500} \]

### Step 5: Calculate the probability of a family having two girls
The number of families with two girls is given as 475. Therefore, the probability of a family having two girls is:
\[ P(\text{two girls}) = \frac{475}{1500} \]

### Step 6: Check if the sum of all probabilities equals 1
We will sum the probabilities calculated in steps 3, 4, and 5:
\[ P(\text{no girls}) + P(\text{one girl}) + P(\text{two girls}) = \frac{211}{1500} + \frac{814}{1500} + \frac{475}{1500} \]
Combining these fractions:
\[ = \frac{211 + 814 + 475}{1500} = \frac{1500}{1500} = 1 \]

### Conclusion
The probability of a family chosen at random having no girls is \( \frac{211}{1500} \), and the sum of all probabilities is indeed 1.

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