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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), the probability of the event E1, is
- a)0
- b)¼
- c)½
- d)1
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
E1 and E2 are events in a probability space satisfying the following c...
let probability of Event E1 = x = prob of E2
prob(E1 union E2) = prob(E1) + prob(E2) - prob(E1 intersect E2)
1 = x + x -x2 (prob(E1 intersect E2) = prob(E1) * prob(E2) as events are independent)
x = 1
prob(E1 union E2) = prob(E1) + prob(E2) - prob(E1 intersect E2)
1 = x + x -x2 (prob(E1 intersect E2) = prob(E1) * prob(E2) as events are independent)
x = 1
Most Upvoted Answer
E1 and E2 are events in a probability space satisfying the following c...
Given Information:
- E1 and E2 are events in a probability space.
- Pr(E1) = Pr(E2)
- Pr(E1 E2) = 1
- E1 and E2 are independent events.
To Find:
The value of Pr(E1), the probability of the event E1.
Solution:
Given that E1 and E2 are independent events, we can use the multiplication rule of probability to find Pr(E1 E2).
The multiplication rule states that for any two events A and B, the probability of their intersection (A B) is equal to the product of their individual probabilities, Pr(A) and Pr(B), if and only if A and B are independent events.
Since E1 and E2 are independent events, we have:
Pr(E1 E2) = Pr(E1) * Pr(E2) ......(1)
Given that Pr(E1 E2) = 1, we can substitute this value in equation (1) to get:
1 = Pr(E1) * Pr(E2)
Since Pr(E1) = Pr(E2), we can rewrite the equation as:
1 = Pr(E1) * Pr(E1)
Simplifying this equation, we get:
1 = (Pr(E1))^2
Taking the square root of both sides, we get:
sqrt(1) = sqrt((Pr(E1))^2)
Therefore, Pr(E1) = 1
Hence, the value of Pr(E1), the probability of the event E1, is 1.
- E1 and E2 are events in a probability space.
- Pr(E1) = Pr(E2)
- Pr(E1 E2) = 1
- E1 and E2 are independent events.
To Find:
The value of Pr(E1), the probability of the event E1.
Solution:
Given that E1 and E2 are independent events, we can use the multiplication rule of probability to find Pr(E1 E2).
The multiplication rule states that for any two events A and B, the probability of their intersection (A B) is equal to the product of their individual probabilities, Pr(A) and Pr(B), if and only if A and B are independent events.
Since E1 and E2 are independent events, we have:
Pr(E1 E2) = Pr(E1) * Pr(E2) ......(1)
Given that Pr(E1 E2) = 1, we can substitute this value in equation (1) to get:
1 = Pr(E1) * Pr(E2)
Since Pr(E1) = Pr(E2), we can rewrite the equation as:
1 = Pr(E1) * Pr(E1)
Simplifying this equation, we get:
1 = (Pr(E1))^2
Taking the square root of both sides, we get:
sqrt(1) = sqrt((Pr(E1))^2)
Therefore, Pr(E1) = 1
Hence, the value of Pr(E1), the probability of the event E1, is 1.
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E1 and E2 are events in a probability space satisfying the following c...
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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 independentThe value of Pr(E1), the probability of the event E1, isa)0b)¼c)½d)1Correct answer is option 'D'. Can you explain this answer?
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