$E_1$ and $E_2$ are events in a probability space satisfying the following constraints: $P(E_1) = P(E_2)$, $P(E_1 \cup E_2) = 1$. If $E_1$ and $E_2$ are independent then find the value of $P(E_1)$.

$E_1$ and $E_2$ are events in a probability space satisfying the following constraints: $P(E_1) = P(E_2)$, $P(E_1 \cup E_2) = 1$. 

If $E_1$ and $E_2$ are independent then $P(E_1) = $ 

(A) 0,    (B) $\dfrac{1}{4}$,    (C) $\dfrac{1}{2}$,    (D) 1 

(GATE 2000) 

Answer: (D) 1  

Explanation: 

Given, $P(E_1) = P(E_2)$, $P(E_1 \cup E_2) = 1$, and $E_1$ and $E_2$ are independent events. 

Hence, $P(E_1 \cap E_2) = P(E_1) \times P(E_2)$. 

Now, $P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1) \times P(E_2)$ 

$\Rightarrow  1 = P(E_2) + P(E_2) - P(E_2) \times P(E_2)$ 

$\Rightarrow  \{P(E_2)\}^2 - 2 \times P(E_2) + 1 = 0$ 

$\Rightarrow  (P(E_2) - 1)^2 = 0$ 

$\Rightarrow  P(E_2) = 1$.