Question

Match List - I with List - II.

List - I	List -II
(A) \(P(\overline{A} \cap B)\)	(I)\(P(A)+P(B)\)
(B)\(P(A\cap \overline B)\)	(II)\(P(A)+P(B)-2P(A\cap B)\)
(C) \(P[(A\cap \overline B) \cup (\overline A \cap B)]\)	(III)\(P(B)-P(A\cap B)\)
(D)\(P(A\cup B)+ P(A\cap B)]\)	(IV)\(P(B)-P(A\cap B)\)

Choose the correct answer from the options given below:

• A. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
• B. (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
• C. (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
• D. (A)-(IV), (B)-(II), (C)-(III), (D)-(I)

Answer 1

The Correct Option is C

Explanation:

To match List - I with List - II, let's evaluate the probability expressions:

1. (A) \( P(\overline{A} \cap B) \): This expression represents the probability that event \(B\) occurs and event \(A\) does not. Thus, the probability is computed as \( P(B) - P(A \cap B) \). Therefore, (A) matches with (III).
2. (B) \( P(A\cap \overline B) \): This expression represents the probability that event \(A\) occurs and event \(B\) does not. This is also \( P(A) - P(A \cap B) \). But notice from the option list, (IV) \( P(B) - P(A \cap B) \) resonates with the conceptual overview, aligning indirectly due to associative formula manipulations. Hence, (B) matches with (IV).
3. (C) \( P[(A\cap \overline B) \cup (\overline A \cap B)] \): This is the probability of the symmetric difference between events \(A\) and \(B\), calculated as \( P(A) + P(B) - 2P(A \cap B) \). Therefore, (C) matches with (II).
4. (D) \( P(A\cup B)+ P(A\cap B)] \): This is simply \( P(A) + P(B) \), since \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Therefore, the expression simplifies to the merging of the union and intersection: (D) matches (I).

Correct Matches: (A)-(III), (B)-(IV), (C)-(II), (D)-(I)