The probability that a student will pass in Mathematics is 3/5 and the probability that he will pass in English is 1/3. If the probability that he will pass in both Mathematics and English is 1/8, what is the probability that he will pass in at least one subject?

Options

A. `97/120`
B. `87/120`
C. `53/120`
D. `120/297`

Answer

To solve the problem, we need to find the probability that a student will pass in at least one subject (Mathematics or English). We can use the formula for the probability of the union of two events.

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- Probability of passing in Mathematics, \( P(M) = \frac{3}{5} \)
- Probability of passing in English, \( P(E) = \frac{1}{3} \)
- Probability of passing in both subjects, \( P(M \cap E) = \frac{1}{8} \)

2. **Use the Formula for the Probability of the Union of Two Events:**
The probability that a student passes in at least one subject is given by:
\[ P(M \cup E) = P(M) + P(E) - P(M \cap E) \]
Here, \( P(M \cup E) \) represents the probability of passing in either Mathematics or English or both.

3. **Substitute the Given Values into the Formula:**
\[ P(M \cup E) = \frac{3}{5} + \frac{1}{3} - \frac{1}{8} \]

4. **Find a Common Denominator:**
The least common multiple (LCM) of the denominators 5, 3, and 8 is 120. We will convert each fraction to have this common denominator:
- Convert \( \frac{3}{5} \):
\[ \frac{3}{5} = \frac{3 \times 24}{5 \times 24} = \frac{72}{120} \]
- Convert \( \frac{1}{3} \):
\[ \frac{1}{3} = \frac{1 \times 40}{3 \times 40} = \frac{40}{120} \]
- Convert \( \frac{1}{8} \):
\[ \frac{1}{8} = \frac{1 \times 15}{8 \times 15} = \frac{15}{120} \]

5. **Substitute Back into the Formula:**
Now substituting back into the equation:
\[ P(M \cup E) = \frac{72}{120} + \frac{40}{120} - \frac{15}{120} \]

6. **Combine the Fractions:**
\[ P(M \cup E) = \frac{72 + 40 - 15}{120} = \frac{97}{120} \]

7. **Final Result:**
The probability that the student will pass in at least one subject is:
\[ P(M \cup E) = \frac{97}{120} \]