If A and B are two events, then which of the following does not represent the probability that exactly one of A, B occurs is

Options

A. `P(A)+P(B)-P(A nnB)`
B. `P(A nnB')+P(A' nn B)`
C. `P(A')+ P(B')-2P(A' nn B')`
D. `P(A)+P(B)-2P(A nn B)`

Answer

To determine which expression does not represent the probability that exactly one of the events A or B occurs, we need to analyze the options provided. The probability that exactly one of the events occurs can be represented mathematically as:

\[ P(A \text{ only}) + P(B \text{ only}) \]

This can be expressed as:

\[ P(A) + P(B) - 2P(A \cap B) \]

Where:
- \( P(A \text{ only}) = P(A) - P(A \cap B) \)
- \( P(B \text{ only}) = P(B) - P(A \cap B) \)

Now, let’s analyze the options step by step.

### Step 1: Analyze Option 1
**Expression:** \( P(A) + P(B) - P(A \cap B) \)

- This expression counts the probability of A and B, but it does not subtract the intersection twice. Thus, it includes the scenario where both A and B occur, which we do not want.
- Therefore, this expression does not represent the probability that exactly one of A or B occurs.

### Step 2: Analyze Option 2
**Expression:** \( P(A \cap B^c) + P(A^c \cap B) \)

- This expression represents the probability of A occurring without B and B occurring without A.
- Hence, it correctly represents the probability that exactly one of A or B occurs.

### Step 3: Analyze Option 3
**Expression:** \( P(A^c) + P(B^c) - 2P(A^c \cap B^c) \)

- This expression represents the probability of neither A nor B occurring, which is not related to the probability of exactly one of A or B occurring.
- Therefore, this expression does not represent the probability that exactly one of A or B occurs.

### Step 4: Analyze Option 4
**Expression:** \( P(A^c \cap B) + P(A \cap B^c) \)

- Similar to Option 2, this expression represents the probability of A occurring without B and B occurring without A.
- Hence, it correctly represents the probability that exactly one of A or B occurs.

### Conclusion
From the analysis, the expression that does not represent the probability that exactly one of A or B occurs is:

**Answer:** \( P(A) + P(B) - P(A \cap B) \)