Two students X and Y appeared in an examination. The probability that X will qualify the examination is 0.05 and Y will qualify the examination is 0.10. The probability that both will qualify the examination is 0.02. What is the probability that only one of them will qualify the examination?

Options

A. 0.15
B. 0.14
C. 0.12
D. 0.11

Answer

To find the probability that only one of the two students, X or Y, qualifies the examination, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- Probability that X qualifies the examination, P(X) = 0.05
- Probability that Y qualifies the examination, P(Y) = 0.10
- Probability that both X and Y qualify the examination, P(X ∩ Y) = 0.02

2. **Understand the Required Probability:**
- We need to find the probability that only one of them qualifies the examination. This can happen in two scenarios:
- Scenario 1: X qualifies and Y does not qualify (X ∩ Y')
- Scenario 2: Y qualifies and X does not qualify (X' ∩ Y)

3. **Calculate the Probability for Each Scenario:**
- For Scenario 1 (X qualifies and Y does not qualify):
\[ P(X \cap Y') = P(X) - P(X \cap Y) \]
Substituting the values:
\[ P(X \cap Y') = 0.05 - 0.02 = 0.03 \]

- For Scenario 2 (Y qualifies and X does not qualify):
\[ P(X' \cap Y) = P(Y) - P(X \cap Y) \]
Substituting the values:
\[ P(X' \cap Y) = 0.10 - 0.02 = 0.08 \]

4. **Combine the Probabilities of Both Scenarios:**
- The total probability that only one of them qualifies is the sum of the probabilities from both scenarios:
\[ P(\text{only one qualifies}) = P(X \cap Y') + P(X' \cap Y) \]
Substituting the values:
\[ P(\text{only one qualifies}) = 0.03 + 0.08 = 0.11 \]

5. **Final Result:**
- Therefore, the probability that only one of them will qualify the examination is **0.11**.