Probability - Dice rolling question [duplicate]

Roll 10 dice. What is the probability the average is between 3 and 4?

I know E[x] = 3.5 for one roll and one would the same for n rolls. I am having difficulty wrapping my head around finding the probability for between the two rolls though.

marked as duplicate by Ross Millikan, Hagen von Eitzen, rschwieb, TMM, EmilyNov 15 '12 at 20:36

• I don't quite know how you are intended to do this. (And additional point: it is not clear whether exactly $3$ or $4$ are meant to be included.) One could use the normal approximation, but $10$ is on the small side for reliability of the approximation. – André Nicolas Nov 15 '12 at 17:52
It sounds like $10$ is large enough not to compute the sum of $10$ random variables. I would opt for the normal approximation, even though as Andre Nicolas correctly notes, it would not be very reliable.
Note that each of your 10 die rolls $X$ is a random variable with expected value of $3.5$ and variance of $E[X^2] - (EX)^2 = 91/6 - 3.5^2 = 35/12$. Using the Central Limit Theorem, we can approximate the sum of these rolls by a normal random variable with mean $35$ and variance $350/12 = 175/6$, and we can compute these in terms of the inverse of the standard normal cdf.