E1 and E2 are events in a probability space satisfying the following…
2000
E1 and E2 are events in a probability space satisfying the following constraints:
Pr(E1) = Pr(E2)
Pr(E1 ∪ E2) = 1
E1 and E2 are independent
The value of Pr(E1) is:
- A.
0
- B.
1/4
- C.
1/2
- D.
1
Attempted by 8 students.
Show answer & explanation
Correct answer: D
The correct answer is: 1.
Let Pr(E1) = Pr(E2) = p.
Since E1 and E2 are independent:
Pr(E1 ∩ E2) = Pr(E1) × Pr(E2) = p².
Using the addition rule:
Pr(E1 ∪ E2) = Pr(E1) + Pr(E2) - Pr(E1 ∩ E2).
Substitute the given condition Pr(E1 ∪ E2) = 1:
1 = p + p - p²
1 = 2p - p²
p² - 2p + 1 = 0
(p - 1)² = 0
p = 1.
Therefore, Pr(E1) = 1.