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 the application 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 theorem 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, find the probability of the union of \(A\) and \(B\):
\[ P(A \cup B) = p_1 + p_2 - p_3 \]
Next, find the probability of the intersection of their complements:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) = 1 - (p_1 + p_2 - p_3) = 1 - p_1 - p_2 + p_3 \]
Now, find the probability of the complement of \(B\) in the denominator:
\[ P(\bar{B}) = 1 - P(B) = 1 - p_2 \]
Substitute these two parts 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).