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A fair die is rolled two times independently. Given that the outcome on the first roll is 1, the expected value of the sum of the two outcomes is
- a)4
- b)4.5
- c)3
- d)5.5
Correct answer is option 'B'. Can you explain this answer?
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A fair die is rolled two times independently. Given that the outcome o...
Let X is a random variable represents an event of rolling a die and P(X = xi) represents the probability of getting sum as xi.
While rolling a die, the probability of getting different numbers as follows.
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A fair die is rolled two times independently. Given that the outcome o...
Solution:
Given that the first roll is 1. Let E be the event that the sum of the two outcomes is x. We need to find E(x), the expected value of the sum of two outcomes.
Let A be the event that the first roll is 1.
The probability of getting 1 on the first roll is 1/6.
The probability of getting x-1 on the second roll is 1/6.
Therefore, P(A∩E) = P(A) * P(E|A) = 1/6 * 1/6 = 1/36.
Now, we need to find the sum of all possible outcomes.
The possible outcomes are (1,1), (1,2), (1,3), (1,4), (1,5), and (1,6).
Their corresponding sums are 2, 3, 4, 5, 6, and 7.
E(x) = ∑x * P(E(x))
= (2 * 1/36) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) + (7 * 1/6)
= 4.5
Therefore, the expected value of the sum of two outcomes is 4.5.
Hence, the correct answer is option 'B'.
Given that the first roll is 1. Let E be the event that the sum of the two outcomes is x. We need to find E(x), the expected value of the sum of two outcomes.
Let A be the event that the first roll is 1.
The probability of getting 1 on the first roll is 1/6.
The probability of getting x-1 on the second roll is 1/6.
Therefore, P(A∩E) = P(A) * P(E|A) = 1/6 * 1/6 = 1/36.
Now, we need to find the sum of all possible outcomes.
The possible outcomes are (1,1), (1,2), (1,3), (1,4), (1,5), and (1,6).
Their corresponding sums are 2, 3, 4, 5, 6, and 7.
E(x) = ∑x * P(E(x))
= (2 * 1/36) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) + (7 * 1/6)
= 4.5
Therefore, the expected value of the sum of two outcomes is 4.5.
Hence, the correct answer is option 'B'.
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A fair die is rolled two times independently. Given that the outcome on the first roll is 1, the expected value of the sum of the two outcomes isa)4b)4.5c)3d)5.5Correct answer is option 'B'. Can you explain this answer?
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