Combinatorics and conditional probability

Question

Assuming seven standard dice are rolled, what is the probability their sum equals 17? Show a general approach to solving this problem analytically, using conditional probability, combinatorics, etc

Answer 1

We have $k=7$ dice, each takes values in $A=\{1,2,3,4,5,6\}$. We want, for $n=17$, to know how many solutions $c_A(n,k)$ of (**)

$$X_1+\cdots+X_k=n$$

there are, where $X_i\in A$; or rather, what the probability of this event is. We can find $c_A(n,k)$ by establishing the identity

$$c_A(n,k)=\sum_{m\in A} c_A(n-m,k-1).$$

(Each solution of (**) must end with $X_k$ taking on one value $m\in A$ and then $X_1+\cdots+X_{k-1}$ must sum to $n-m$ (note, for the initial conditions, that $c_A(n,1)=1$ iff $n\in A$)). Then, the probability $P[X_1+\cdots+X_k=n]$ is

$$\frac{c_A(n,k)}{6^k}$$

since there are in total $6^k$ values that the variables $X_1,\ldots,X_k$ can take on. Plugging in numbers, we find that your desired probability is

$$\frac{6538}{279936}=2.33...\%$$