In a college 30% students fail in physics, 25% fail in mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she failed in mathematics is

Options

A. `(1)/(10)`
B. `(2)/(5)`
C. `(9)/(20)`
D. None of these

Answer

To solve the problem, we need to find the probability that a student fails in Physics given that she has failed in Mathematics. We will use the concept of conditional probability for this.

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- Probability of failing in Physics, \( P(P) = 30\% = \frac{30}{100} = \frac{3}{10} \)
- Probability of failing in Mathematics, \( P(M) = 25\% = \frac{25}{100} = \frac{1}{4} \)
- Probability of failing in both Physics and Mathematics, \( P(P \cap M) = 10\% = \frac{10}{100} = \frac{1}{10} \)

2. **Apply the Formula for Conditional Probability:**
We want to find \( P(P | M) \), the probability that a student fails in Physics given that she has failed in Mathematics. The formula for conditional probability is:
\[ P(P | M) = \frac{P(P \cap M)}{P(M)} \]

3. **Substitute the Values:**
- From step 1, we have:
- \( P(P \cap M) = \frac{1}{10} \)
- \( P(M) = \frac{1}{4} \)
- Now substituting these values into the formula:
\[ P(P | M) = \frac{\frac{1}{10}}{\frac{1}{4}} \]

4. **Simplify the Expression:**
- Dividing fractions involves multiplying by the reciprocal:
\[ P(P | M) = \frac{1}{10} \times \frac{4}{1} = \frac{4}{10} = \frac{2}{5} \]

5. **Conclusion:**
The probability that a student fails in Physics given that she has failed in Mathematics is \( \frac{2}{5} \).