Correct : 4.25

To find the expected value of a discrete random variable X, we use the standard formula for expected value (which is also the mean of the distribution):
E[X] = ∑ [x × P(X = x)]
This means we must multiply each possible value of X by its corresponding probability, and then add all those products together.
1. Group the values by their probabilities
We are given two different probability values in this problem:
- Values with probability 1/6: {1, 2, 5, 7}
- Values with probability 1/12: {3, 4, 6, 8}
2. Calculate the sum for the first group (Probability = 1/6)
Sum₁ = 1(1/6) + 2(1/6) + 5(1/6) + 7(1/6)
Sum₁ = (1 + 2 + 5 + 7) / 6
Sum₁ = 15 / 6 = 2.5
3. Calculate the sum for the second group (Probability = 1/12)
Sum₂ = 3(1/12) + 4(1/12) + 6(1/12) + 8(1/12)
Sum₂ = (3 + 4 + 6 + 8) / 12
Sum₂ = 21 / 12 = 7 / 4 = 1.75
4. Calculate the Total Expected Value E[X]
E[X] = Sum₁ + Sum₂
E[X] = 2.5 + 1.75 = 4.25
Correct Answer: 4.25