Probability for total score after rolling a die $n$ times

Question

I'm trying to solve the following problem:

Say, you roll a fair die 10 times, and call $Y$ the total number of points scored. What's the approximate probability that $Y$ lies between 35 and 45(extreme included)?

I computed the mean first: $E[S_{10}]=E[X_1 + X_2 + ... +X_n]=nE[X_i]=10 \times 3.5 =35$

For the variance, I computed the second moment first of a single roll:

$E[X_i^2]=\sum_{1}^{6} i^2 P(X_i=i)=91/6$

And the variance of the single roll yields:

$Var[X_i]=E[X_i^2]-E[X_i]^2=\frac{91}{6}-3.5^2$

Then, because of the independence assumption: $Var[S_{10}]=n Var[X_i]= 29.17$.

What about the last point? Is it correct to assume $S_n\sim N(35,29.17)$ because of the central limit theorem (but $n=10$ is small), and then compute the CDF accordingly?

Answer 1

Yes, you have the right idea. The Normal distribution gives a good approximation to the total roll of ten dice, as shown in the figure below. The dots are the pdf of the dice total (computed by use of a generating function), and the smooth curve is the Normal approximation.