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:

Linearity of expectation, \(E(A+B) = E(A) + E(B)\), holds absolutely regardless of the dependency between variables. Thus, simply multiplying the mean of one trial by the number of trials provides the fastest answer.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the expected value (mean) of the sum of the numbers appearing on three fair dice when they are thrown.
Step 2: Key Formula or Approach:
By the property of linearity of expectation, the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent: \[ E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3) \] Step 3: Detailed Explanation:
Let \(X_i\) be the outcome of the \(i\)-th die. The possible outcomes are \(\{1, 2, 3, 4, 5, 6\}\), each with a probability of \(\frac{1}{6}\).
The expected value for a single die roll is: \[ E(X_i) = \sum x \cdot P(x) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \] Since three dice are thrown, the expected sum is: \[ E(X_1 + X_2 + X_3) = 3.5 + 3.5 + 3.5 = 10.5 \] Step 4: Final Answer:
The mean of the sum of the numbers is \(10.5\).