If A and B are two events, then which one of the following is/are always true?

Options

A. `P(AnnB)geP(A)+P(B)-1`
B. `P(AnnB)leP(A)`
C. `P(A'nnB')geP(A')+P(B')-1`
D. `P(AnnB)=P(A)P(B)`

Explanation

To determine which of the given statements about events A and B are always true, we will analyze each option step by step.

### Step 1: Understanding the Union and Intersection of Events
We start with the fundamental relationship between the probabilities of two events A and B:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This relationship will help us analyze the options.

### Step 2: Analyzing Option 1
**Option 1:** \( P(A \cap B) \geq P(A) + P(B) - 1 \)

From the formula for the union of two events, we can rearrange it:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
Since \( P(A \cup B) \) cannot exceed 1, we have:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]
This statement is always true.

### Step 3: Analyzing Option 2
**Option 2:** \( P(A \cap B) \leq P(A) \)

This statement is also true because the intersection of A and B cannot exceed the probability of A alone. Thus, \( P(A \cap B) \) is always less than or equal to \( P(A) \).

### Step 4: Analyzing Option 3
**Option 3:** \( P(A') \cap B' = 1 - P(A) \)

This option is a bit misleading. The complement of A (denoted as \( A' \)) and the complement of B (denoted as \( B' \)) do not have a straightforward relationship with the probabilities of A and B. However, we can use the formula:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Thus, this statement is not always true.

### Step 5: Analyzing Option 4
**Option 4:** \( P(A \cap B) = P(A) \cdot P(B) \)

This statement is only true if A and B are independent events. Since independence is not guaranteed, this statement is not always true.

### Conclusion
After analyzing all the options, we conclude that:
- **Options 1 and 2 are always true.**
- **Options 3 and 4 are not always true.**

### Final Answer
The statements that are always true are:
1. \( P(A \cap B) \geq P(A) + P(B) - 1 \)
2. \( P(A \cap B) \leq P(A) \)