Question
Download Solution PDFFor two events A and B, P(A) = P(A|B) = 0.25 and P(BIA) = 0.5. Which of the following are correct?
I. A and B are independent.
II. P(Ac ∪ Bc) = 0.875
III. P(Ac ∩ Bc) = 0.375
Select the answer using the code given below.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
Given:
\(P(A) = P(\frac{A}{B}) = 0.25\)
and \(P(\frac{B}{A}) = 0.5\)
I. \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(A)}\)
⇒ P(A∩B) = P(A) P(B|A)
⇒ P(A∩B) = 0.25 × 0.5 = 0.125
Now
⇒ \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(B)}\)
⇒ \(P(B)= \frac{P(A∩ B)}{P(\frac{A}{B})}\)
⇒ \(P(B) = \frac{0.125}{0.25} = 0.5\)
Now, P(A).P(B) = 0.25 × 0.5 = 0.125 = P(A∩B)
Thus A and B are independent
II. \(P(\overline A\cup \overline B ) = 1 – P(A ∩ B)\)
= 1 – 0.125 = 0.875
III. \(P(\overline A∩ \overline B ) = 1 – P(A \cup B)\)
= 1 – [P(A) + P(B) – P(A ∩ B)
= 1 – [0.25 + 0.5 – 0.125]
= = 1 – 0.625 = 0.375
So all statements I, II, and III are correct.
∴ Option (d) is correct.
Last updated on May 30, 2025
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