JEE Main 2020 · 8 January · morning · Single correct · +4 / −1

Let A and B be two independent events such that
P(A) = 1/3 and P(B) = 1/6.
Then, which of the following is TRUE?

  1. (a) P( A/(A ∪ B) ) = 1/4
  2. (b) P( A/B ) = 2/3
  3. (c) P( A'/B' ) = 1/3
  4. (d) P( A/B' ) = 1/3
Answer and solution

Answer (d) P( A/B' ) = 1/3

Solution

Maths sir · 3 steps

Independence means conditioning on B or B' changes nothing, so P(A|B') is just P(A).

  1. Use independence. When A and B are independent, so are A and B', A' and B, and A' and B'. So P(A | B) = P(A | B') = P(A) = 1/3 and P(A' | B') = P(A') = 2/3.
  2. Find the union. P(A ∩ B) = 1/3 × 1/6 = 1/18, so P(A ∪ B) = 1/3 + 1/6 - 1/18 = 8/18 = 4/9. Since A sits inside A ∪ B, their intersection is just A.

    P(A | A ∪ B) = P(A)/P(A ∪ B) = (1/3)/(4/9) = 3/4

    OptionActual valueClaimedVerdict
    (a) P(A | A ∪ B)3/41/4false
    (b) P(A | B)P(A) = 1/32/3false
    (c) P(A' | B')P(A') = 2/31/3false
    (d) P(A | B')P(A) = 1/31/3true
  3. So. Only (d) holds: P(A | B') = P(A) = 1/3.

The idea to keep. If A and B are independent, every pair from A, A', B, B' is independent too. Conditioning on any of them leaves the other probability unchanged.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Probability — Advanced

Asked about 100 times in JEE Main since 2020. This question is from 2020.