If `P(A cup B) = P(A) + P(B)`, then what can be said about the events `A` and `B` ?

Answer

To solve the problem, we need to analyze the given equation and apply the properties of probability.

### Step-by-step Solution:

1. **Understanding the Given Information**:
We are given that:
\[ P(A \cup B) = P(A) + P(B) \]
This means the probability of the union of events A and B is equal to the sum of their individual probabilities.

2. **Using the General Formula for Probability of Union**:
The general formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Here, \( P(A \cap B) \) represents the probability of both events A and B occurring simultaneously.

3. **Setting the Two Equations Equal**:
Since we have:
\[ P(A \cup B) = P(A) + P(B) \]
We can substitute the general formula into this equation:
\[ P(A) + P(B) - P(A \cap B) = P(A) + P(B) \]

4. **Simplifying the Equation**:
By simplifying the equation, we can cancel \( P(A) + P(B) \) from both sides:
\[ -P(A \cap B) = 0 \]
This implies:
\[ P(A \cap B) = 0 \]

5. **Interpreting the Result**:
The result \( P(A \cap B) = 0 \) indicates that the probability of both events A and B occurring together is zero. This means that there are no outcomes that are common to both events.

6. **Conclusion**:
Since \( P(A \cap B) = 0 \), we can conclude that events A and B are **mutually exclusive** or **disjoint** events. This means that the occurrence of one event excludes the occurrence of the other.

### Final Answer:
Events A and B are mutually exclusive (or disjoint).

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