**Given: **E_{1} and E_{2} are two events such that P(E_{1}) = \(\frac{1}{4}\) and P(E_{2}) = \(\frac{1}{3}\) and

P(E_{1} \(\cup\) E_{2}) = \(\frac{1}{2}\)

**To show:** E_{1} and E_{2} are independent events.

**We know that,**

Hence, P(E_{1} ∩ E_{2}) = = P(E_{1}) + P(E_{2}) - P(E_{1} \(\cup\) E_{2})

= \(\frac{1}{4}+\frac{1}{3}-\frac{1}{2}\)

= \(\frac{1}{12}\) Equation 1

Since The condition for two events to be independent is

P(E_{1} ∩ E_{2}) = P(E_{1}) x P(E_{2})

= \(\frac{1}{4}\times \frac{1}{3}\)

= \(\frac{1}{12}\) Equation 2

**Since, **Equation 1 = Equation 2

\(\Rightarrow\) E_{1} and E_{2} are independent events.

**Hence proved.**