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If two events A and B are independent, then
- a)A and the complement of B are independent
- b)B and the complement of A are independent
- c)Complements of A and B are independent
- d)All of these (a), (b) and (c)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If two events A and B are independent, thena)A and the complement of B...
Events A and B are independent if the equation P(A∩B) = P(A) P(B) holds true. You can use the equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.
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If two events A and B are independent, thena)A and the complement of B...
Explanation:
When two events A and B are independent, it means that the occurrence of one event does not affect the occurrence of the other event. In other words, the probability of one event happening does not change based on whether or not the other event has happened.
(a) A and the complement of B are independent:
The complement of B is the event that B does not occur. It is denoted by B'. If A and B are independent, then P(A) = P(A|B') because the occurrence of B' does not affect the probability of A. Therefore, A and B' are also independent.
(b) B and the complement of A are independent:
Similarly, if A and B are independent, then P(B) = P(B|A') because the occurrence of A' does not affect the probability of B. Therefore, B and A' are also independent.
(c) Complements of A and B are independent:
If A and B are independent, then P(A') = 1 - P(A) and P(B') = 1 - P(B). Using the formula for independent events, we have P(A and B') = P(A)P(B') and P(A' and B) = P(A')P(B). Substituting the values of P(A') and P(B') in the first equation and simplifying, we get P(A and B') = (1 - P(B))P(A). Similarly, substituting the values of P(A') and P(B) in the second equation and simplifying, we get P(A' and B) = (1 - P(A))P(B). Therefore, P(A and B') = P(A)P(B') and P(A' and B) = P(A')P(B), which shows that the complements of A and B are also independent.
(d) All of these (a), (b) and (c):
From the above explanations, we can conclude that all three options (a), (b), and (c) are correct. Therefore, the correct answer is option (d).
When two events A and B are independent, it means that the occurrence of one event does not affect the occurrence of the other event. In other words, the probability of one event happening does not change based on whether or not the other event has happened.
(a) A and the complement of B are independent:
The complement of B is the event that B does not occur. It is denoted by B'. If A and B are independent, then P(A) = P(A|B') because the occurrence of B' does not affect the probability of A. Therefore, A and B' are also independent.
(b) B and the complement of A are independent:
Similarly, if A and B are independent, then P(B) = P(B|A') because the occurrence of A' does not affect the probability of B. Therefore, B and A' are also independent.
(c) Complements of A and B are independent:
If A and B are independent, then P(A') = 1 - P(A) and P(B') = 1 - P(B). Using the formula for independent events, we have P(A and B') = P(A)P(B') and P(A' and B) = P(A')P(B). Substituting the values of P(A') and P(B') in the first equation and simplifying, we get P(A and B') = (1 - P(B))P(A). Similarly, substituting the values of P(A') and P(B) in the second equation and simplifying, we get P(A' and B) = (1 - P(A))P(B). Therefore, P(A and B') = P(A)P(B') and P(A' and B) = P(A')P(B), which shows that the complements of A and B are also independent.
(d) All of these (a), (b) and (c):
From the above explanations, we can conclude that all three options (a), (b), and (c) are correct. Therefore, the correct answer is option (d).
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If two events A and B are independent, thena)A and the complement of B are independentb)B and the complement of A are independentc)Complements of A and B are independentd)All of these (a), (b) and (c)Correct answer is option 'D'. Can you explain this answer?
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