Expected Sum of Ten Dice

Write the total as a sum of ten independent die values.

Each die has expected value $3.5$.

1. 1. Expected value of one die

   A single fair six-sided die has mean $\frac{1+2+\cdots +6}{6}=3.5$.

2. 2. Sum via linearity

   By linearity of expectation, the total expected sum is ten times a single die's mean.

   $$E[\text{sum}]=10\times 3.5=35$$

Answer: 35