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If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these?
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If two event A and B are independent events then (a) They can not be e...
Understanding Independent Events
Independent events A and B are defined by the property that the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this is expressed as:
- P(A ∩ B) = P(A) * P(B)
This property has implications for the relationships between independent events and other types of events.
Exclusive Events
- **Definition**: Two events are exclusive (or mutually exclusive) if they cannot occur simultaneously. For example, if A occurs, B cannot occur.
- **Relationship with Independence**: If A and B are independent, then:
- P(A ∩ B) = 0 (because they cannot happen together)
- However, since P(A ∩ B) = P(A) * P(B) for independent events, this leads to a contradiction unless either P(A) = 0 or P(B) = 0.
Thus, independent events cannot be exclusive.
Exhaustive Events
- **Definition**: Events A and B are exhaustive if they cover all possible outcomes in a sample space.
- **Relationship with Independence**: Independent events can indeed be exhaustive. For example, if A is the event of rolling an even number on a die, and B is the event of rolling an odd number, these are independent and exhaustive.
Conclusion
Based on the definitions and relationships:
- (a) They cannot be exclusive and vice versa: **True**
- (b) They can be exhaustive events and vice versa: **True**
- (c) They can be exclusive but not exhaustive: **False**
Therefore, the correct answer is (a).
Independent events A and B are defined by the property that the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this is expressed as:
- P(A ∩ B) = P(A) * P(B)
This property has implications for the relationships between independent events and other types of events.
Exclusive Events
- **Definition**: Two events are exclusive (or mutually exclusive) if they cannot occur simultaneously. For example, if A occurs, B cannot occur.
- **Relationship with Independence**: If A and B are independent, then:
- P(A ∩ B) = 0 (because they cannot happen together)
- However, since P(A ∩ B) = P(A) * P(B) for independent events, this leads to a contradiction unless either P(A) = 0 or P(B) = 0.
Thus, independent events cannot be exclusive.
Exhaustive Events
- **Definition**: Events A and B are exhaustive if they cover all possible outcomes in a sample space.
- **Relationship with Independence**: Independent events can indeed be exhaustive. For example, if A is the event of rolling an even number on a die, and B is the event of rolling an odd number, these are independent and exhaustive.
Conclusion
Based on the definitions and relationships:
- (a) They cannot be exclusive and vice versa: **True**
- (b) They can be exhaustive events and vice versa: **True**
- (c) They can be exclusive but not exhaustive: **False**
Therefore, the correct answer is (a).
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If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these?
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If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these?.
If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two event A and B are independent events then (a) They can not be exclusive and vice – versa (b) They can be exhaustive events and vice – versa (c) They can be exclusive but not exhaustive (d) None of these?.
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