Question

Let A and B are any two events and P(A)= p\(_1\) , P(B)= p\(_2\) and P(A \(\cap\) B)= p\(_3\). What is the value of P\(\left(\frac{\bar{A}}{\bar{B}}\right)\) ?

• A. \(\frac{1- \text{p}_1 - \text{p}_2 + \text{p}_3}{1- \text{p}_3}\)
• B. \(\frac{\text{p}_1}{\text{p}_2}\)
• C. \(\frac{1- \text{p}_1 - \text{p}_2 + \text{p}_3}{1- \text{p}_2}\)
• D. \(\frac{1- \text{p}_1 - \text{p}_2 + \text{p}_3}{1- \text{p}_1}\)

Hint:

To solve set-based probability questions quickly:
Remember that the numerator of \(P(\bar{A} \mid \bar{B})\) is the region outside the union, which is always:
\[ 1 - \text{Union} = 1 - P(A) - P(B) + \text{Intersection} \]
The denominator is always the complement of the given condition:
\[ 1 - P(B) \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires applying the laws of set theory, De Morgan's laws, and the definition of conditional probability.
We want to find the conditional probability of the complement of \(A\) given the complement of \(B\).
Key Formula or Approach:
The conditional probability formula is:
\[ P\left(\frac{\bar{A}}{\bar{B}}\right) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})} \]
By De Morgan's Law:
\[ \bar{A} \cap \bar{B} = \overline{A \cup B} \implies P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
The addition rule of probability is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Step 2: Detailed Explanation:
Let us substitute the given values into our equations:
- \(P(A) = p_1\)
- \(P(B) = p_2\)
- \(P(A \cap B) = p_3\)
First, express the union of the two events using the addition rule:
\[ P(A \cup B) = p_1 + p_2 - p_3 \]
Next, find the probability of the intersection of the complements using De Morgan's Law:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) = 1 - (p_1 + p_2 - p_3) = 1 - p_1 - p_2 + p_3 \]
Now, calculate the probability of the conditioning event \(\bar{B}\):
\[ P(\bar{B}) = 1 - P(B) = 1 - p_2 \]
Finally, substitute these components back into the conditional probability formula:
\[ P\left(\frac{\bar{A}}{\bar{B}}\right) = \frac{1 - p_1 - p_2 + p_3}{1 - p_2} \]
This matches Option (C).

Step 3: Final Answer:
The correct option is (C).