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

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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.