Which sum shows up most often when you roll 10 dice?
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The sum that shows up most often when you roll 10 dice is 35.
Are you also interested to see a reason why this is so?
The key to this question is the concept of unimodality (q.v.).
Try to prove by induction that if you take the sum of $n \geq 2$ cubes then the distribution has the following form:
If you find the proof difficult, try again with two-sided dice.
Of course, as $n \rightarrow \infty$ the distribution will approach (in some sense) a Gaussian, but I'm not sure this is enough to prove anything specific about the mode (other than its asymptotics).
For each individual die, its 'expected value' is 7/2. This is because it can be either 1, 2, 3, 4, 5 or 6, each with a probability of 1/6 (assuming it's a fair die). If you want to know what value will come up most often for 1 die, this is the sum of each of it's possible values times the probability of that value. Which means 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 1/6(1 + 2 + 3 + 4 + 5 + 6) = 7/2. Obviously, 3.5 will never show up on a die, and this makes sense because if each side has equal probability, you shouldn't expect to see one side more than the others.
To find the expected value of multiple die, you can simply add up each individual expected value. For all 10 dice, this value is 7/2, so you have 7/2 * 10 = 35
Thus, the expected value of rolling 10 dice is 35.
Wikipedia has more information on probability and expected values, if it interests you: