If A and B be two mutually exclusive events in a sample space such that , `P(A) = 2/5 and P(B) = 1/2 ` then
find `P ( barA nn B )`:

Options

A. `1/2`
B. `3/5`
C. `4/7`
D. `7/15`

Answer

To solve the problem, we need to find the probability of the complement of the intersection of two mutually exclusive events A and B. Let's break it down step by step.

### Step 1: Understand Mutually Exclusive Events
Mutually exclusive events are events that cannot occur at the same time. This means that the probability of both events A and B occurring simultaneously is zero:
\[ P(A \cap B) = 0 \]

### Step 2: Given Probabilities
We are given:
\[ P(A) = \frac{2}{5} \]
\[ P(B) = \frac{1}{2} \]

### Step 3: Find the Probability of the Intersection
Since A and B are mutually exclusive, the probability of their intersection is:
\[ P(A \cap B) = 0 \]

### Step 4: Find the Probability of the Complement of the Intersection
We need to find \( P(\bar{A} \cap B) \). The complement of the intersection can be expressed as:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]

### Step 5: Substitute the Values
Now substituting the values we have:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]
\[ P(\bar{A} \cap B) = \frac{1}{2} - 0 \]
\[ P(\bar{A} \cap B) = \frac{1}{2} \]

### Final Answer
Thus, the probability of the complement of A intersection B is:
\[ P(\bar{A} \cap B) = \frac{1}{2} \]

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