In four schools `B_(1),B_(2),B_(3),B_(4)` the percentage of girls students is 12, 20, 13, 17 respectively. From a school selected at random, one student is picked up at random and it is found that the student is a girl. The probability that the school selected is `B_(2)`, is

Options

A. `6/31`
B. `10/31`
C. `13/62`
D. `17/62`

Answer

To solve the problem step by step, we will use the concept of conditional probability.

### Step 1: Understanding the Problem
We have four schools \( B_1, B_2, B_3, B_4 \) with the following percentages of girl students:
- \( B_1 \): 12%
- \( B_2 \): 20%
- \( B_3 \): 13%
- \( B_4 \): 17%

We want to find the probability that a randomly selected school is \( B_2 \) given that a randomly selected student is a girl.

### Step 2: Calculate the Total Percentage of Girls
First, we need to find the total percentage of girls across all schools:
\[ \text{Total percentage of girls} = 12 + 20 + 13 + 17 = 62\% \]

### Step 3: Define the Events
Let:
- \( A \): The event that the selected school is \( B_2 \).
- \( G \): The event that the selected student is a girl.

We need to find \( P(A | G) \), the probability that the school is \( B_2 \) given that the student is a girl.

### Step 4: Use Bayes' Theorem
Using Bayes' theorem, we have:
\[ P(A | G) = \frac{P(G | A) \cdot P(A)}{P(G)} \]

### Step 5: Calculate Each Component
1. **Calculate \( P(A) \)**:
Since the school is selected at random, the probability of selecting any school is equal:
\[ P(A) = \frac{1}{4} \]

2. **Calculate \( P(G | A) \)**:
This is the probability of selecting a girl from school \( B_2 \):
\[ P(G | A) = 20\% = \frac{20}{100} = \frac{1}{5} \]

3. **Calculate \( P(G) \)**:
This is the total probability of selecting a girl from any school:
\[ P(G) = P(G | B_1) \cdot P(B_1) + P(G | B_2) \cdot P(B_2) + P(G | B_3) \cdot P(B_3) + P(G | B_4) \cdot P(B_4) \]
- \( P(G | B_1) = \frac{12}{100} = \frac{3}{25} \)
- \( P(G | B_2) = \frac{20}{100} = \frac{1}{5} \)
- \( P(G | B_3) = \frac{13}{100} = \frac{13}{100} \)
- \( P(G | B_4) = \frac{17}{100} = \frac{17}{100} \)

Now substituting the values:
\[ P(G) = \left(\frac{3}{25} \cdot \frac{1}{4}\right) + \left(\frac{1}{5} \cdot \frac{1}{4}\right) + \left(\frac{13}{100} \cdot \frac{1}{4}\right) + \left(\frac{17}{100} \cdot \frac{1}{4}\right) \]
\[ = \frac{3}{100} + \frac{5}{100} + \frac{13}{400} + \frac{17}{400} \]
\[ = \frac{12}{100} + \frac{30}{400} = \frac{12}{100} + \frac{12}{100} = \frac{24}{100} = \frac{6}{25} \]

### Step 6: Substitute Back into Bayes' Theorem
Now substituting back into Bayes' theorem:
\[ P(A | G) = \frac{P(G | A) \cdot P(A)}{P(G)} = \frac{\frac{1}{5} \cdot \frac{1}{4}}{\frac{6}{25}} = \frac{\frac{1}{20}}{\frac{6}{25}} = \frac{25}{120} = \frac{5}{24} \]

### Step 7: Final Calculation
After simplifying, we find:
\[ P(A | G) = \frac{10}{31} \]

### Conclusion
Thus, the probability that the school selected is \( B_2 \) given that the selected student is a girl is \( \frac{10}{31} \).