Question

If three dice are thrown, then the mean of the sum of the numbers appearing on them is

• A. 58.5
• B. 76.66
• C. 71.75
• D. 10.5

Hint:

The mean outcome of a fair die numbered 1 to \( n \) is \( \frac{n+1}{2} \). For a standard die, it is 3.5.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We need to find the expected value (mean) of the sum of outcomes when three dice are thrown. The expectation of a sum is the sum of the expectations.
Step 2: Key Formula or Approach:

\[ E(S) = E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3) \] where \( X_i \) is the outcome of the \( i \)-th die.
Step 3: Detailed Explanation:

The expected value of a single die roll \( X \) is: \[ E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \] Since there are 3 independent dice: \[ \text{Mean Sum} = 3 \times 3.5 = 10.5 \]
Step 4: Final Answer:

The mean is 10.5.