What is the expected value of a fair six-sided die?

What is the expected value of a fair six-sided die?

• A. 3
• B. 3.25
• C. 3.75
• D. 3.5

Answer: (D)

Think of the expected value as the long-run average when you roll the die many times. For a fair six-sided die, each face from 1 to 6 is equally likely, with probability 1/6. So the expected value is the sum of each outcome times its probability: (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5. An intuitive check is that the outcomes are symmetric around 3.5: pairs (1,6), (2,5), (3,4) each average to 3.5, pulling the overall average to 3.5. Thus the expected value is 3.5.

Think of the expected value as the long-run average when you roll the die many times. For a fair six-sided die, each face from 1 to 6 is equally likely, with probability 1/6. So the expected value is the sum of each outcome times its probability: (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5.

An intuitive check is that the outcomes are symmetric around 3.5: pairs (1,6), (2,5), (3,4) each average to 3.5, pulling the overall average to 3.5. Thus the expected value is 3.5.