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 two independent events E and F with probabilities:

\[ P(E) = \frac{1}{4}, \quad P(E \cup F) = \frac{2}{5}, \quad P(F) = a \]

Step 1: Use the formula for probability of union of two independent events:

\[ P(E \cup F) = P(E) + P(F) - P(E)P(F) \] \[ \frac{2}{5} = \frac{1}{4} + a - \frac{1}{4}a \]

Step 2: Solve for \( a \):

\[ \frac{2}{5} - \frac{1}{4} = a - \frac{1}{4}a \] \[ \frac{8}{20} - \frac{5}{20} = \frac{3}{4}a \] \[ \frac{3}{20} = \frac{3}{4}a \] \[ a = \frac{3}{20} \times \frac{4}{3} = \frac{12}{60} = \frac{1}{5} \]

The correct value of \( a \) is \( \frac{1}{5} \).

Answer 2

Given 

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

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 \('a’\)

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

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

\(⇒\dfrac{-3a}{4}=\dfrac{-3}{20}\)

\(⇒a=\dfrac{1}{5}\)