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NDA 2 2024 Maths, question 115

Topic: Probability

Q115

For two events AA and BB, P(A)=P(A∣B)=0.25P(A)=P(A|B)=0.25 and P(B∣A)=0.5P(B|A)=0.5. Which of the following are correct?

I. AA and BB are independent.

II. P(Ac∪Bc)=0.875P(A^{c}\cup B^{c})=0.875

III. P(Ac∩Bc)=0.375P(A^{c}\cap B^{c})=0.375

Select the answer using the code given below.

  1. (a) I and II only
  2. (b) II and III only
  3. (c) I and III only
  4. (d) I, II and IIICorrect answer

Solution.

Answer: (d) I, II and III

  1. P(A∩B)=P(B|A)P(A)=0.5×0.25=0.125.
  2. Since P(A|B)=0.25, P(B)=0.125/0.25=0.5, and P(A)P(B)=0.125, proving independence.
  3. De Morgan gives P(Aᶜ∪Bᶜ)=1−0.125=0.875 and P(Aᶜ∩Bᶜ)=1−(0.25+0.5−0.125)=0.375.
  4. All statements are correct.