Probability · Class 12 · CBSE · Science Mathematics
If events A and B are independent, show that A and B' are independent and that A' and B' are also independent.
Practice question from Probability (Class 12 · CBSE · Science Mathematics), with the correct answer and a step-by-step solution.
hard · Subjective · Q-8DFD02
If events A and B are independent, show that A and B' are independent and that A' and B' are also independent.
Answer: Since A and B are independent, P(A ∩ B) = P(A)P(B). Thus, P(A ∩ B') = P(A) − P(A ∩ B) = P(A)[1 − P(B)] = P(A)P(B'), so A and B' are independent. Also, P(A' ∩ B') = 1 − P(A ∪ B) = 1 − P(A) − P(B) + P(A)P(B) = P(A')P(B'), so A' and B' are independent.
Solution
Use the definition of independence, , and the complement identities. The resulting products equal and , respectively.
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