Question

S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B. 

 
Then the correct match is:

• A. I - e, II - d, III - c, IV - b
• B. I - a, II - c, III - b, IV - d
• C. I - d, II - c, III - b, IV - a
• D. I - b, II - d, III - a, IV - e

Hint:

When working with probabilities, remember that mutually exclusive events cannot occur at the same time, while independent events have a product relationship for their intersection. The union of events A and B can be calculated based on their relationship.

Answer 1

The Correct Option is C

Let's break down each option in List A and match it with the appropriate option in List B. I. A, B are mutually exclusive events:
For mutually exclusive events, the occurrence of one event excludes the occurrence of the other event. Thus, the probability of their union is simply the sum of the probabilities of the individual events: \[ P(A \cup B) = P(A) + P(B). \] So, the correct match for this is (IV) from List B. II. A, B are independent events:
For independent events, the occurrence of one event does not affect the occurrence of the other. The probability of the intersection of two independent events is the product of their individual probabilities: \[ P(A \cap B) = P(A)P(B). \] Thus, the correct match for this is (d) from List B. III. A \cap B = A:
If \( A \cap B = A \), this means that event \( A \) completely occurs within event \( B \). In such cases, the probability of \( A \cup B \) will simply be the probability of \( B \): \[ P(A \cup B) = P(B). \] So, the correct match for this is (c) from List B. IV. A \cup B = S:
If \( A \cup B = S \), this means that the union of events \( A \) and \( B \) covers the entire sample space. The probability of their union would be 1: \[ P(A \cup B) = 1. \] Thus, the correct match for this is (a) from List B. ### Final Answer: - I matches with (d) from List B. - II matches with (c) from List B. - III matches with (b) from List B. - IV matches with (a) from List B. Thus, the correct match is: \[ \boxed{I - d, II - c, III - b, IV - a}. \]

Answer 2

Understanding the Matching of Events
Let's analyze each statement from List A and correctly match it with its corresponding expression from List B based on fundamental probability rules.

I. A, B are mutually exclusive events:
Mutually exclusive events cannot occur together, i.e.,
\[ A \cap B = \emptyset \]
Therefore, the probability of their union is:
\[ P(A \cup B) = P(A) + P(B) \]
Correct Match: (d) from List B.

II. A, B are independent events:
Independence implies that the occurrence of one does not affect the other. So:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Correct Match: (c) from List B.

III. \( A \cap B = A \):
This means that event A is completely contained within B (A ⊆ B). Then:
\[ P(A \cup B) = P(B) \]
Correct Match: (b) from List B.

IV. \( A \cup B = S \):
The union of A and B covers the entire sample space, so:
\[ P(A \cup B) = 1 \]
Correct Match: (a) from List B.

Final Matching:
I → (d), II → (c), III → (b), IV → (a)

Correct Answer:
\[ \boxed{I - d, \quad II - c, \quad III - b, \quad IV - a} \]