"If " `P(AcupB)=P(AcapB)`" for any two events A and B, then"`

Options

A. `P(A)=P(B)`
B. `P(A)gtP(B)`
C. `P(A)ltP(B)`
D. None of these

Explanation

To solve the problem "If \( P(A \cup B) = P(A \cap B) \) for any two events A and B, then", we will follow these steps:

### Step 1: Understand the given equation
We start with the equation:
\[ P(A \cup B) = P(A \cap B) \]
This means that the probability of the union of events A and B is equal to the probability of their intersection.

### Step 2: Use the formula for the probability of the union
We know the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting this into our equation gives:
\[ P(A) + P(B) - P(A \cap B) = P(A \cap B) \]

### Step 3: Rearrange the equation
Rearranging the equation, we have:
\[ P(A) + P(B) = 2P(A \cap B) \]

### Step 4: Analyze the implications
From the equation \( P(A) + P(B) = 2P(A \cap B) \), we can see that both \( P(A) \) and \( P(B) \) must be equal to \( P(A \cap B) \) for the equation to hold true. This leads us to conclude:
\[ P(A) = P(A \cap B) \quad \text{and} \quad P(B) = P(A \cap B) \]

### Step 5: Conclude the relationship between A and B
Since \( P(A) = P(A \cap B) \), it implies that event A occurs only when event B occurs. Similarly, \( P(B) = P(A \cap B) \) implies that event B occurs only when event A occurs. Therefore, we conclude that:
\[ P(A) = P(B) \]
and events A and B must be equal in terms of their probabilities.

### Final Conclusion
Thus, we can conclude that if \( P(A \cup B) = P(A \cap B) \) for any two events A and B, then:
\[ P(A) = P(B) \]