# Expanding a probability generating function for the total score of rolling a die r times

An unbiased six-faced die is rolled r times. The probability generating function for the total score is

$$[\frac{t(1-t^6)}{6(1-t)}]^r$$

Hence show that the probability of the total score being (r+3) is

$$\frac{1}{6}^{r+1}r(r+1)(r+2)$$

I appreciate any help anyone can provide and this is a statistics question in a Further Maths A-Level style paper for context on how they want the question to be solved.

## 2 Answers

Let $$[t^n] f(t)$$ denote the coefficient of $$t^n$$ in $$f(t)$$. (I'm not sure how standard this notation is but it's very useful.) There is a factor of $$t^r$$ in the generating function.

$$[t^{r+3}] \left( t (1-t^6) \over 6(1-t) \right)^r = [t^3] \left( 1-t^6 \over 6(1-t) \right)^r$$

and you can pull out a factor of $$(1/6)^r$$ as well to get

$$[t^3] \left( {1-t^6 \over 1-t} \right)^r$$

At this point there are two possible methods.

(1) Note that

$$\left( {1-t^6 \over 1-t} \right)^r = (1 + t + t^2 + t^3 + t^4 + t^5)^r$$

and think about how multiplication of polynomials works to get the coefficient of $$t^3$$ - you can get a factor of $$t^3$$ from one factor of $$t^3$$ and the rest 1, or a $$t^2$$ and a $$t$$, or three copies of $$t$$. (This isn't all that different from solving the problem without generating functions.)

(2) Take the third derivative and evaluate at zero. Since you have an $$r$$ in the exponent this might be a bit tricky - I'd try logarithmic differentiation. I have not actually done this.

• thanks for you help, I ended up using your first method and using permutations and combinations to find out how many ways you can get $t^3$ and got the right answer. I think method 2 is beyond my knowledge but thank you anyway! – Dev Vadhwana Feb 20 '19 at 16:30
• You're welcome. I'm only slightly familiar with the UK system so I wasn't sure what was reasonable to assume in the Further Maths context. – Michael Lugo Feb 20 '19 at 16:55

Picking up from where Michael Lugo left off, we want to compute $$[t^3]{(1-t^6)^r \over (1-t)^r}.$$ Expand the numerator via the binomial theorem and drop any terms that involve a power of $$t$$ greater than $$3$$: $$[t^3]{(1-t^6)^r \over (1-t)^r} = [t^3]{1\over (1-t)^r}.$$ Now use the identity $${1\over(1-s)^{k+1}} = \sum_{n=0}^\infty \binom{n+k}k s^n$$ to get $$[t^3]{1\over (1-t)^r} = \binom{r+2}{r-1} = \binom{r+2}3 = {r(r+1)(r+2)\over6}.$$ Multiply this by the factor of $$1/6^r$$ that was pulled out earlier to get the final result.

• thanks for continuing the answer. this is a much more concise method than the one i ended up using so thanks for sharing this method! – Dev Vadhwana Feb 25 '19 at 20:11