If A and B are two events, the probability that exactly one of them occurs is given by

Options

A. `P(A) + P(B) - 2P(A nn B)`
B. `P(A nn barB) + P(barA nn B)`
C. `P(A uu B) - P(A nn B)`
D. `P(barA) + P(barB) - 2P(barA nn barB)`

Explanation

To find the probability that exactly one of the two events A and B occurs, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Events**:
- We have two events, A and B.
- We want to find the probability that exactly one of these events occurs.

2. **Using Set Notation**:
- The event that exactly one of A or B occurs can be expressed as:
\[ (A \cap B') \cup (A' \cap B) \]
- Here, \(A'\) is the complement of A (i.e., A does not occur), and \(B'\) is the complement of B.

3. **Applying the Probability Rule**:
- The probability of the union of two events can be calculated using:
\[ P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \]
- Therefore, we can write:
\[ P((A \cap B') \cup (A' \cap B)) = P(A \cap B') + P(A' \cap B) \]

4. **Calculating Individual Probabilities**:
- We can express \(P(A \cap B')\) and \(P(A' \cap B)\) as follows:
- \(P(A \cap B') = P(A) - P(A \cap B)\)
- \(P(A' \cap B) = P(B) - P(A \cap B)\)

5. **Combining the Results**:
- Substitute these expressions back into the equation:
\[ P((A \cap B') \cup (A' \cap B)) = (P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) \]
- Simplifying this gives:
\[ P(A) + P(B) - 2P(A \cap B) \]

6. **Final Result**:
- Therefore, the probability that exactly one of the events A or B occurs is:
\[ P(A) + P(B) - 2P(A \cap B) \]