Expected value of the max of $2$ dice What is the expected value of the max of two dice?
I just wonder if there's a better way to get the answer to this question than listing out all possible outcomes and determining the expected value from there as this is actually an interview question?
Thanks 

Suppose the dices have $n$ faces. Let $X=\max(a,b)$
Note that:
$X=1$ only in the case (1,1), i.e. $p_X(1)=1/n^2$
$X=2$ in the cases (1,2), (2,2), (2,1), i.e. $p_X(2)=3/n^2$
$\vdots$
$X=k$ in $k+k-1=2k-1$ cases, i.e. $p_X(k)=(2k-1)/n^2$(this is easy to see when you draw all the cases).
$$E(X)=\frac{1}{n^2} \sum_{x=1}^{n}x(2x-1)=\frac{1}{n^2} \sum_{x=1}^{n}(2x^2-1)=\\ \frac{1}{n^2}\left(\frac{n(n+1)(2n+1)}{3}-\frac{n(n+1)}{2}\right)=\frac{4n^3+3n^2-n}{6n^2} = \dfrac{4n^2 + 3n - 1}{6n}$$
