If A and B are two events such that P(A) = 0.5, P(B) = 0.6 and P(A`uu` B) = 0.8. Find `P(A/B)`.

Options

A. `1/3`
B. `1/2`
C. `1/4`
D. none of these

Answer

To solve the problem step by step, we will use the information given and apply the relevant formulas in probability.

### Step 1: Write down the given probabilities
We are given:
- \( P(A) = 0.5 \)
- \( P(B) = 0.6 \)
- \( P(A \cup B) = 0.8 \)

### Step 2: Use the formula for the probability of the union of two events
The formula for the probability of the union of two events \( A \) and \( B \) is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula, we have:
\[ 0.8 = 0.5 + 0.6 - P(A \cap B) \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 0.8 = 1.1 - P(A \cap B) \]

### Step 5: Solve for \( P(A \cap B) \)
Rearranging the equation gives:
\[ P(A \cap B) = 1.1 - 0.8 \]
\[ P(A \cap B) = 0.3 \]

### Step 6: Find the conditional probability \( P(A | B) \)
The formula for conditional probability is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the values we have:
\[ P(A | B) = \frac{0.3}{0.6} \]

### Step 7: Simplify the fraction
Now, simplify the fraction:
\[ P(A | B) = \frac{1}{2} \]

### Final Answer
Thus, the probability \( P(A | B) \) is \( \frac{1}{2} \).

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