Suppose A and B are two events such that P(A) = 0.5 and P(B) = 0.8, then which one of the following is not true?

Options

A. `P(AnnB) le 0.5`
B. `P(AnnB) ge 0.3`
C. `P(A' nn B ) le 0.5`
D. `P(A nn B') le 0.1`

Answer

To solve the problem, we need to analyze the probabilities of events A and B, and determine which of the statements regarding these probabilities is not true. We are given:

- \( P(A) = 0.5 \)
- \( P(B) = 0.8 \)

We will evaluate the statements one by one.

### Step 1: Understand the basic properties of probabilities
The probability of the union of two events A and B can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This means that the probability of either A or B occurring is equal to the sum of their individual probabilities minus the probability of both A and B occurring.

### Step 2: Calculate the maximum possible value of \( P(A \cap B) \)
Since the probability of any event cannot exceed 1, we can derive:
\[ P(A \cup B) \leq 1 \]
Substituting the values we have:
\[ P(A) + P(B) - P(A \cap B) \leq 1 \]
\[ 0.5 + 0.8 - P(A \cap B) \leq 1 \]
\[ 1.3 - P(A \cap B) \leq 1 \]
\[ P(A \cap B) \geq 0.3 \]
This means that the minimum value of \( P(A \cap B) \) is 0.3.

### Step 3: Calculate the maximum possible value of \( P(A \cap B) \)
Since \( P(A) = 0.5 \), the maximum value of \( P(A \cap B) \) cannot exceed \( P(A) \):
\[ P(A \cap B) \leq P(A) = 0.5 \]

### Step 4: Analyze the statements
Now we will analyze the statements provided in the question:

1. **Statement 1**: \( P(A \cap B) \leq 0.5 \)
This is true since \( P(A \cap B) \) cannot exceed \( P(A) \).

2. **Statement 2**: \( P(A \cap B) > 0.3 \)
This is not necessarily true since \( P(A \cap B) \) could be exactly 0.3.

3. **Statement 3**: \( P(A^c \cap B) \leq 0.5 \)
This is true because \( P(A^c \cap B) = P(B) - P(A \cap B) \) and since \( P(A \cap B) \geq 0.3 \), \( P(A^c \cap B) \) will be less than or equal to 0.5.

4. **Statement 4**: \( P(A \cap B^c) < 0.2 \)
This is true since \( P(A \cap B^c) = P(A) - P(A \cap B) \) and since \( P(A \cap B) \) can be at most 0.5, \( P(A \cap B^c) \) can be less than 0.2.

### Conclusion
The statement that is not true is **Statement 2**: \( P(A \cap B) > 0.3 \).