Given Constraints:
1. Pr(E1) = Pr(E2)
2. Pr( E1 U E2) = 1
3. E1 and E2 are independent
As we know: Pr(E1 U E2) = Pr(E1) + Pr(E2) – Pr(E1 ∩ E2) As E1 and E2 are independent events. (cond.3)
So Pr(E1 ∩ E2) = Pr(E1) Pr(E2) Pr(E1) = Pr(E2) (cond.2)
let probability of Event E1 = x = prob of E2 So, Pr(E1 U E2) = Pr(E1) + Pr(E2) – Pr(E1) Pr(E2) 1 = x + x -x* x (cond. 1) 1=2x-x^2 x^2-2x+1 = 0 (x-1)^2 = 0 x = 1 So, Pr(E1) = Pr(E2) = 1 Thus, option (D) is the answer.
Reference : https://people.richland.edu/james/lecture/m170/ch05-rul.html
Another Solution : E1 and E2 are independent events.
Pr(E1 U E2) = Pr(E1) + Pr(E2) – Pr(E1) Pr(E2)
Pr(E1) = Pr(E2) (given)
So,
2 * Pr(E1) – Pr(E1)^{2} = Pr( E1 U E2)
2 * Pr(E1) – Pr(E1)^{2} = 1
So, Pr(E1) = Pr(E2) = 1
Thus, option (D) is the answer.