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Probability that the sum of all values of 5 pairs of dice will be between 30 and 40

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.

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

This question was marked as an exact duplicate of an existing question.

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.

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