Let A and B be two events such that `P(A cap B')=0.20,P(A'cap B)=0.15, P(A'cap B')=0.1`, then `p(A//B)` is equal to ,

Options

A. `11//14`
B. `2//11`
C. `2//7`
D. `1//7`.

Answer

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

1. **Given Information:**
- \( P(A \cap B') = 0.20 \)
- \( P(A' \cap B) = 0.15 \)
- \( P(A' \cap B') = 0.10 \)

2. **Understanding the Events:**
- \( P(A \cap B') \) represents the probability of event A occurring while event B does not occur.
- \( P(A' \cap B) \) represents the probability of event B occurring while event A does not occur.
- \( P(A' \cap B') \) represents the probability of neither event A nor event B occurring.

3. **Finding \( P(A \cap B) \):**
- The total probability must equal 1:
\[ P(A \cap B) + P(A \cap B') + P(A' \cap B) + P(A' \cap B') = 1 \]
- Substituting the known values:
\[ P(A \cap B) + 0.20 + 0.15 + 0.10 = 1 \]
- Simplifying:
\[ P(A \cap B) + 0.45 = 1 \]
- Thus,
\[ P(A \cap B) = 1 - 0.45 = 0.55 \]

4. **Finding \( P(B) \):**
- We can find \( P(B) \) using the relationship:
\[ P(B) = P(A \cap B) + P(A' \cap B) \]
- Substituting the known values:
\[ P(B) = 0.55 + 0.15 = 0.70 \]

5. **Calculating \( P(A | B) \):**
- The formula for conditional probability is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
- Substituting the values we found:
\[ P(A | B) = \frac{0.55}{0.70} \]
- Simplifying:
\[ P(A | B) = \frac{55}{70} = \frac{11}{14} \]

6. **Final Answer:**
- Therefore, \( P(A | B) = \frac{11}{14} \).

---