If P(not A) = 0.7, P(B) = 0.7 and `P((B)/(A)) =` 0. 5 then find P `((A)/(B)) and P (A cup B) `

Answer

To solve the problem, we need to find \( P(A|B) \) and \( P(A \cup B) \) given the following probabilities:

- \( P(\neg A) = 0.7 \)
- \( P(B) = 0.7 \)
- \( P(B|A) = 0.5 \)

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

We know that:
\[ P(A) = 1 - P(\neg A) \]
Substituting the given value:
\[ P(A) = 1 - 0.7 = 0.3 \]

### Step 2: Find \( P(A \cap B) \)

We can use the formula for conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Rearranging gives us:
\[ P(A \cap B) = P(B|A) \times P(A) \]
Now substituting the known values:
\[ P(A \cap B) = 0.5 \times 0.3 = 0.15 \]

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

Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we have:
\[ P(A \cup B) = 0.3 + 0.7 - 0.15 \]
Calculating this gives:
\[ P(A \cup B) = 1.0 - 0.15 = 0.85 \]

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

Using the formula for conditional probability again:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ P(A|B) = \frac{0.15}{0.7} \]
Calculating this gives:
\[ P(A|B) = \frac{15}{70} = \frac{3}{14} \approx 0.2143 \]

### Final Results

Thus, we have:
- \( P(A|B) \approx 0.2143 \)
- \( P(A \cup B) = 0.85 \)