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If E1 and E2 are the two events such that P(E1) = 1/4, P(E2) = 1/3 and P(E1 \(\cup\) E2) = 1/ , show that E1 and Eare independent events.

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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.

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