A and B are two events such that odds against A are 2:1 odds in favour of A ∪ B are 3 : 1. If x ≤ P(B) ≤ y, then the ordered pair (x,y) is
Answer & explanation
Correct answer: option 1
We have,
$P(A)=\frac{1}{3} $ and $ P(A ∪ B)=\frac{3}{4}$
$∴ P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$
$⇒ \frac{3}{4}=\frac{1}{3}+P(B) -P(A ∩ B)$
$⇒ \frac{5}{12}=P(B)-P(A ∩ B)$
$⇒ P(B) =\frac{5}{12}+P(A ∩ B) ⇒ P(B) ≥ 5/12$ .....(i)
Again,
$P(B)=\frac{5}{12} + P(A ∩ B)$
$⇒ P(B) ≤ 5/12 + P(A)$ $[∵P(A∩B) ≤ P(A)]$
$⇒ P(B) ≤ \frac{5}{12}+\frac{1}{3}=\frac{3}{4}$ .......(ii)
From (i) and (ii), we obtain $ 5/12 ≤ P(B) ≤ 3/4 $
Hence, $ x = 5/12 $ and $ y = 3/4$