For my forthcoming book Math Games with Bad Drawings, I considered including a classic dice game called “Drop Dead.” It narrowly missed the cut. Strike one: the lousy name. Strike two: it’s a game of pure chance, with no room for decision-making. And strike three: well, that name again.
Still, I want to share the game here, because it teaches a useful lesson about the mathematics of risk.
On your turn, you roll five dice, and score their sum, with one big exception: 2’s and 5’s are fatal. They immediately drop dead and are removed from play. Not only that, but whenever any 2’s or 5’s appear, the other dice are worthless; you score no points for the roll. Then, whether you scored points or not, you roll all remaining dice again, and continue repeating this process until all five dice have dropped dead. Play for a set number of turns per player (say four), after which the highest total score wins.
Here’s a sample turns that lasted for six rolls, scoring a total of 15 points.
Notice that on my first roll, I didn’t score any points. Die #4 (by coming up with a two) negated the collective efforts of Die #1, Die #2, Die #3, and Die #5.
That kind of failure is common. You’ll score on your opening roll just 13% of the time. A single 2 or 5 suffices to spoil the party, and with five potential party spoilers, few parties remain unspoiled. You thus wind up scoring most of your points with just one or two dice remaining, because smaller “parties” are more likely to go off successfully.
This leads to our larger theme, and the lesson that interests me: Starting with extra dice barely helps. In fact, there’s little benefit past your eighth die, and almost none past your twelfth.
It seems weird. An extra die can’t hurt, can it? Best-case scenario, it adds to your score, and worst-case scenario, it comes up 2 or 5, at which point you discard it, and wind up right back where you started.
Well, sure, it can’t hurt. But past a certain point, it doesn’t much help, either. Each die has a 1-in-3 risk of dropping dead. Compounded many times, that becomes a virtual guarantee: somebody is going to spoil the party. With just twenty dice, the probability of entirely avoiding 2’s and 5’s is just 0.03%, roughly your lifetime chance of being struck by lightning.
Let’s say you begin with 5 quadrillion dice, enough to blanket the state of West Virginia. Seems like you should score tons of points, right? Nope. Roll after roll, about 1/3 of your dice will spoil the party. This will repeat a hundred times in succession, your score stuck on zero, until finally, with just a few dice remaining, you begin to score points. (About 17 points, on average.)
5,000,000,000,000,000 dice. 17 points.
The moral: Don’t design systems where everything needs to go right. If your machine is doomed by one broken part; if your party is spoiled by one late guest; if your game plan crumbles when one player strays out of position; then you’ve got yourself a problem. A lot of problems, actually: one per component. Crowds are good for some things, but achieving unanimity is not one of them.
By the way, if you want to turn this into an actual game, Joe Kisenwether has a good idea: You may start with as many dice as you want, but your turn ends immediately after your 5th roll. Thus, you want to pick enough dice that you don’t run out (1 or 2 is probably too few) but not so many that you waste early rolls on scoring zero (so 20 is too many).
Puzzle: in this version, what’s the optimal number of dice?