GATE | GATE-CS-2000 | Question 25

E1 and E2 are events in a probability space satisfying the following constraints:

 
    Pr(E1) = Pr(E2)
    Pr(EI U E2) = 1
    E1 and E2 are independent 

The value of Pr(E1), the probability of the event E1 is

(A) 0
(B) 1/4
(C) 1/2
(D) 1


Answer: (D)

Explanation:



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

This solution is contributed by Nitika Bansal.

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.

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