Given: E1 and E2 are two events such that P(E1) = \(\frac{1}{4}\) and P(E2) = \(\frac{1}{3}\) and
P(E1 \(\cup\) E2) = \(\frac{1}{2}\)
To show: E1 and E2 are independent events.
We know that,
Hence, P(E1 ∩ E2) = = P(E1) + P(E2) - P(E1 \(\cup\) E2)
= \(\frac{1}{4}+\frac{1}{3}-\frac{1}{2}\)
= \(\frac{1}{12}\) Equation 1
Since The condition for two events to be independent is
P(E1 ∩ E2) = P(E1) x P(E2)
= \(\frac{1}{4}\times \frac{1}{3}\)
= \(\frac{1}{12}\) Equation 2
Since, Equation 1 = Equation 2
\(\Rightarrow\) E1 and E2 are independent events.
Hence proved.