Question:
If three dice are thrown, then the mean of the sum of the numbers appearing on them is
If three dice are thrown, then the mean of the sum of the numbers appearing on them is
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The mean outcome of a fair die numbered 1 to \( n \) is \( \frac{n+1}{2} \). For a standard die, it is 3.5.
Updated On: Mar 30, 2026
- 58.5
- 76.66
- 71.75
- 10.5
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The Correct Option is D
Solution and Explanation
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.
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.
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