Question

Consider two independent events \(E\) and \(F\) such that\( P(E)=1/4,P(E∪F)=2/5\) and \(P(F)=a\).Then,the value of \(a\) is

• A. \(\dfrac{13}{20}\)
• B. \(\dfrac{1}{20}\)
• C. \(\dfrac{1}{4}\)
• D. \(\dfrac{1}{5}\)
• E. \(\dfrac{3}{5}\)

Answer 1

The Correct Option is D

Given Data

\( P(E)=\dfrac{1}{4},P(E∪F)=\dfrac{2}{5}\)

We know that,

\(P(E ∪ F) = P(E) + P(F) - P(E ∩ F)\)

\(P(E ∩ F) = \dfrac{1}{4} + a - \dfrac{2}{5}\)---------(1)

Now, \(P(E ∩ F) = P(E) × P(F)\)

\(P(E ∩ F) = (1/4) × a\)----------(2)

Now, we can equate (1) and (2) to get the value of \('a’\)

\(\dfrac{1}{4} × a = \dfrac{1}{4} + a - \dfrac{2}{5}\)

\(⇒\dfrac{1}{4} × a -a = \dfrac{-3}{20}\)

\(⇒\dfrac{-3a}{4}=\dfrac{-3}{20}\)

\(⇒a=\dfrac{1}{5}\) (_Ans)