Misc 19 - If P(A) + P(B) - P(A and B) = P(A), then P(B|A) - Miscellaneous

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Misc 19 If A and B are any two events such that P(A) + P(B) – P(A and B) = P(A), then (A) P(B|A) = 1 (B) P(A|B) = 1 (C) P(B|A) = 0 (D) P(A|B) = 0 Here P(A and B) = P(A ∩ B) Now, P(A) + P(B) – P(A and B) = P(A) P(A) + P(B) – P(A ∩ B) = P(A) P(B) – P(A ∩ B) = P(A) – P(A) P(B) – P(A ∩ B) = 0 P(B) = P(A ∩ B) P(B) = P(A ∩ B) Dividing By P(B) 1 = 𝑃(𝐴 ∩ 𝐵)﷮𝑃(𝐵)﷯ 𝑃(𝐴 ∩ 𝐵)﷮𝑃(𝐵)﷯ = 1 Also, P(A|B) = 𝑃(𝐴 ∩ 𝐵)﷮𝑃(𝐵)﷯ So, P(A|B) = 1 Hence, Option (B) is Correct.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.