Expected Value of the Lower of Two Dice

The mirror of the higher-of-two case, and derived the same way. The smaller of two dice is at least $k$ exactly when both dice are at least $k$, which happens with probability $((d-k+1)/d)^{2}$. Sum those over all $k$ — a standard trick for expectations of non-negative whole numbers — and the result collapses to $(d+1)(2d+1)/(6d)$.

On six-sided dice that is $91/36$, about 2.53, against 3.5 for a single die. On a twenty-sided die it is 7.175 against 10.5. The penalty is the same size as the bonus on the higher-of-two page, and that is not a coincidence: the two averages are symmetric about the single-die average, so a disadvantage costs precisely what an advantage gains. That is a genuinely useful fact for anyone comparing two rules that look asymmetric, because it means they are not.

The sum $E_{\min} + E_{\max} = d + 1$ is the cleanest check available on either formula, and it is worth doing every time. On a twenty-sided die, $7.175 + 13.825 = 21.000$ exactly. The argument is the one given on the companion page: sorting two dice into a larger and a smaller does not change what they are, so the two expectations must add to twice the average of one die. If your version of either formula fails this, the error is in that formula and not in the check.

There is a small asymmetry in the algebra that is worth noticing if you invert either relation. Solving the maximum backwards for the number of faces gives a quadratic with one positive root, so it is a clean inverse. Solving the minimum backwards gives a quadratic whose two roots multiply to one half — so the second root is always below one, never a real die, and the physical answer is again unique. The two formulas behave differently under inversion despite being reflections of one another, which is the sort of thing that only shows up when you try it.

And the standing caution about independence applies here too. Two dice do not coordinate. A low result on one says nothing about the other, and no sequence of past pairs changes what the next pair will do.