`P(A cup B)=P(A cap B)` if and only if the relation between `P(A) and P(B)` is …………

Explanation

To solve the problem, we need to analyze the condition given: \( P(A \cup B) = P(A \cap B) \).

### Step 1: Understand the Definitions
Recall the definitions of union and intersection in probability:
- \( P(A \cup B) \) is the probability that either event A occurs, event B occurs, or both occur.
- \( P(A \cap B) \) is the probability that both events A and B occur simultaneously.

### Step 2: Use the Formula for Union
We can use the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Set Up the Equation
Given that \( P(A \cup B) = P(A \cap B) \), we can set up the equation:
\[ P(A) + P(B) - P(A \cap B) = P(A \cap B) \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ P(A) + P(B) = 2P(A \cap B) \]

### Step 5: Analyze the Condition
If we denote \( P(A) \) as \( p_A \) and \( P(B) \) as \( p_B \), we can express the above equation as:
\[ p_A + p_B = 2P(A \cap B) \]
This suggests that the sum of the probabilities of A and B is twice the probability of their intersection.

### Step 6: Consider the Case When A and B are Equal
The only way for \( P(A \cup B) \) to equal \( P(A \cap B) \) is if A and B are the same event. In this case:
\[ P(A) = P(B) = P(A \cap B) \]

### Conclusion
Thus, the relation between \( P(A) \) and \( P(B) \) is that they must be equal:
\[ P(A) = P(B) \]

### Final Answer
The relation between \( P(A) \) and \( P(B) \) is that \( P(A) = P(B) \).

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