Statistics – Probability Sum of 5 Pairs of Dice Values Between 30 and 40

dicenormal distributionstatistics

I'm trying to solve a question that asks:

If 5 pairs of fair dice are rolled, approximate the probability that the sum
of the values obtained is between 30 and 40 inclusive.

My approach so far, was to create a random variable called $T$ whose range is from $10$ to $60$ and look for
$$P\{|T-35| \le 5\} $$

I don't know what to do next. Can I proceed like this is a normal distribution or is five too small a sample to qualify for the Central Limit Theorem?

Best Answer

This can be determined exactly with the use of generating functions (also known as probability generating functions) and a computer algebra system (like GP/PARI).

The probability is the sum of the coefficients of the terms in the polynomial $$ \left(\frac{1}{6}x+\frac{1}{6}x^2+\frac{1}{6}x^3+\frac{1}{6}x^4+\frac{1}{6}x^5+\frac{1}{6}x^6\right)^{10} $$ with degree 30 through 40, which is the sum of the coefficients of the terms in the polynomial $$ (x+x^2+x^3+x^4+x^5+x^6)^{10} $$ with degree 30 through 40, divided by $6^{10}$.

Using GP, this is calculated instantly as

sum(i=30,40,polcoeff(sum(j=1,6,x^j)^10,i))/(6^10)

which gives the result $\frac{10384949}{15116544} \approx 0.68699227812918084$.

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