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

Options

A. `1/10`
B. `2/5`
C. `9/20`
D. `1/3`

Answer

To solve the problem step by step, we will use the concept of conditional probability. Let's denote the events as follows:

- Let \( A \) be the event that a student fails in Physics.
- Let \( B \) be the event that a student fails in Mathematics.

### Step 1: Identify the given probabilities
From the problem, we have the following information:
- \( P(A) = 30\% = \frac{30}{100} = 0.3 \) (Probability of failing in Physics)
- \( P(B) = 25\% = \frac{25}{100} = 0.25 \) (Probability of failing in Mathematics)
- \( P(A \cap B) = 10\% = \frac{10}{100} = 0.1 \) (Probability of failing in both subjects)

### Step 2: Use the formula for conditional probability
We need to find the probability that a student fails in Physics given that she has failed in Mathematics, which is denoted as \( P(A | B) \). The formula for conditional probability is:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

### Step 3: Substitute the known values into the formula
Now we will substitute the values we identified in Step 1 into the formula:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.25} \]

### Step 4: Calculate the probability
Now we will perform the division:

\[ P(A | B) = \frac{0.1}{0.25} = \frac{10}{25} = \frac{2}{5} \]

### Conclusion
Thus, the probability that a student fails in Physics given that she has failed in Mathematics is:

\[ P(A | B) = \frac{2}{5} \]