Consider the following in respect of two events A and B.
I. P(A occurs but not B) = P(A) - P(B) if `C sub A`
II.P(A alone or B alone occurs) = P(A) + P(B) - `P(A nn B)`
III `P(A uu B) = P(A) + P(B) if A and B are mutually exclusive.
Which of the above is/are correct?

Options

A. 1 only
B. I and III only
C. 2 and 3 only
D. 1 and 2 only

Answer

To solve the problem, we need to analyze each of the three statements regarding the probabilities of events A and B.

### Step-by-step Solution:

1. **Statement I: P(A occurs but not B) = P(A) - P(B) if C is a subset of A.**
- The probability of A occurring but not B can be expressed as:
\[ P(A \text{ and not } B) = P(A) - P(A \cap B) \]
- If C is a subset of A, we cannot directly say that \( P(A \text{ and not } B) = P(A) - P(B) \). This is incorrect because it does not account for the intersection of A and B correctly. Thus, **Statement I is incorrect.**

2. **Statement II: P(A alone or B alone occurs) = P(A) + P(B) - P(A ∩ B).**
- The probability of A alone or B alone occurring can be expressed as:
\[ P(A \text{ alone}) + P(B \text{ alone}) = P(A) - P(A \cap B) + P(B) - P(A \cap B) = P(A) + P(B) - 2P(A \cap B) \]
- The statement claims \( P(A \text{ alone or } B \text{ alone}) = P(A) + P(B) - P(A \cap B) \), which is incorrect because it does not account for the double counting of \( P(A \cap B) \). Thus, **Statement II is incorrect.**

3. **Statement III: P(A ∪ B) = P(A) + P(B) if A and B are mutually exclusive.**
- If A and B are mutually exclusive, then:
\[ P(A \cap B) = 0 \]
- Therefore, the probability of A or B occurring is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - 0 = P(A) + P(B) \]
- Thus, **Statement III is correct.**

### Conclusion:
Only Statement III is correct.