If `P(A)=0.6,P(B)=0.7` and `P(A uu B)=0.9`, then
(i) `P(A//B)`=_______ (ii) `P(B//A)` =________

Answer

To solve the problem, we will use the formulas for conditional probability and the relationship between union and intersection of events.

Given:
- \( P(A) = 0.6 \)
- \( P(B) = 0.7 \)
- \( P(A \cup B) = 0.9 \)

We need to find:
1. \( P(A|B) \)
2. \( P(B|A) \)

### Step 1: Find \( P(A \cap B) \)

We can use the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values:
\[ 0.9 = 0.6 + 0.7 - P(A \cap B) \]

### Step 2: Solve for \( P(A \cap B) \)

Rearranging the equation:
\[ P(A \cap B) = 0.6 + 0.7 - 0.9 \]
\[ P(A \cap B) = 1.3 - 0.9 = 0.4 \]

### Step 3: Find \( P(A|B) \)

Using the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Substituting the values we have:
\[ P(A|B) = \frac{0.4}{0.7} \]

### Step 4: Simplify \( P(A|B) \)

Calculating the fraction:
\[ P(A|B) = \frac{4}{7} \]

### Step 5: Find \( P(B|A) \)

Using the formula for conditional probability again:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the values:
\[ P(B|A) = \frac{0.4}{0.6} \]

### Step 6: Simplify \( P(B|A) \)

Calculating the fraction:
\[ P(B|A) = \frac{4}{6} = \frac{2}{3} \]

### Final Answers:
(i) \( P(A|B) = \frac{4}{7} \)
(ii) \( P(B|A) = \frac{2}{3} \)

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