For two mutually exclusive events A and B. P(A) = 0.2 and `P(barA nn B)=0.3`. What is `P (A|(A uu B))` equal to?

Options

A. `(1)/(2)`
B. `(2)/(5)`
C. `(2)/(7)`
D. `(2)/(3)`

Answer

To solve the problem, we need to find the conditional probability \( P(A | (A \cup B)) \).

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = 0.2 \)
- \( P(\bar{A} \cap B) = 0.3 \)

### Step 2: Calculate \( P(B) \)
Since \( P(\bar{A} \cap B) \) represents the probability of event B occurring while A does not occur, we can express this as:
\[ P(B) = P(\bar{A} \cap B) + P(A \cap B) \]
Since A and B are mutually exclusive, \( P(A \cap B) = 0 \). Therefore:
\[ P(B) = P(\bar{A} \cap B) = 0.3 \]

### Step 3: Calculate \( P(A \cup B) \)
Using the formula for the probability of the union of two mutually exclusive events:
\[ P(A \cup B) = P(A) + P(B) \]
Substituting the values we have:
\[ P(A \cup B) = 0.2 + 0.3 = 0.5 \]

### Step 4: Calculate \( P(A | (A \cup B)) \)
The conditional probability \( P(A | (A \cup B)) \) is given by the formula:
\[ P(A | (A \cup B)) = \frac{P(A \cap (A \cup B))}{P(A \cup B)} \]
Since \( A \cap (A \cup B) = A \), we can simplify this to:
\[ P(A | (A \cup B)) = \frac{P(A)}{P(A \cup B)} \]
Substituting the values we calculated:
\[ P(A | (A \cup B)) = \frac{0.2}{0.5} = 0.4 \]

### Final Answer
Thus, the value of \( P(A | (A \cup B)) \) is \( 0.4 \).