If A and B are events of the same experiments with `P(A) = 0.2, P(B) = 0.5,` the maximum value of `P(barA nn B)` is

Options

A. `1//4`
B. `1//2`
C. `1//8`
D. `1//16`

Answer

To find the maximum value of \( P(\bar{A} \cap B) \), we will follow these steps:

### Step 1: Understand the Given Probabilities
We are given:
- \( P(A) = 0.2 \)
- \( P(B) = 0.5 \)

### Step 2: Calculate the Probability of the Complement of A
The probability of the complement of A, denoted as \( P(\bar{A}) \), is calculated as:
\[ P(\bar{A}) = 1 - P(A) = 1 - 0.2 = 0.8 \]

### Step 3: Identify the Relationship Between Events
We want to find the maximum value of \( P(\bar{A} \cap B) \). To do this, we need to consider the relationship between \( P(\bar{A}) \) and \( P(B) \).

### Step 4: Determine the Maximum Value of \( P(\bar{A} \cap B) \)
The maximum value of \( P(\bar{A} \cap B) \) occurs when \( \bar{A} \) and \( B \) overlap as much as possible. However, since \( P(\bar{A}) = 0.8 \) and \( P(B) = 0.5 \), the maximum overlap cannot exceed the probability of \( B \).

Thus, the maximum value of \( P(\bar{A} \cap B) \) is limited by \( P(B) \):
\[ P(\bar{A} \cap B) \leq P(B) = 0.5 \]

### Step 5: Conclusion
Therefore, the maximum value of \( P(\bar{A} \cap B) \) is:
\[ \text{Maximum value of } P(\bar{A} \cap B) = 0.5 \]

### Final Answer
The maximum value of \( P(\bar{A} \cap B) \) is \( 0.5 \).

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