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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Last Updated :
28 Jun, 2021
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