pre-rolling the dice

The new episode of Darths and Droids – Ãœberstition – has this truly inspired meta-commentary at the bottom, which purports to quantify the dice superstition that all RPG gamers suffer from to varying degree:

Pete, being the highly logical, calculating person he is, rejects all of that as superstitious nonsense. He instead applies the scientific approach. Over the years, he’s collected somewhere around a thousand twenty-sided dice. Every so often, he gathers them all together. He sits down at a table and carefully and individually rolls each of the thousand dice, once. Of course, roughly a twentieth of them will roll a one. He takes those fifty-odd dice and rolls them a second time. After about an hour of concentrated dice rolling, he’ll end up with around two or three dice that have rolled two ones in a row. He takes those primed dice and places them in special custom-made padded containers where they can’t roll around, and carries them to all the games he plays.

Then, when in the most dire circumstances, where a roll of one would be absolutely disastrous, he pulls out the prepared dice. He now has in his hand a die that has rolled two ones in a row. Pete knows the odds of a d20 rolling three ones in a row is a puny one in 8,000. He has effectively pre-rolled the ones out of the die, and can make his crucial roll with confidence.

This is the sort of geek brilliance that you’d normally find over at XKCD (though this forum thread comes close).

Being the geek that I am, and also because I just sent in a draft of a paper so the ball isn’t in my court and I can goof off a bit, I wonder if we cant look at Pete’s empirical superstition more critically. First, we can write a script in MATLAB to actually implement Pete’s strategy and see whether the empirical results match expectation. Second, we can analyze the problem theoretically.

I’ll play with MATLAB later – as far as the theory goes, though, Pete is out of luck. Each die roll is a purely independent event, so the probability of rolling a 1 is always 1 in 20. Pete argues that the special dice have already rolled 1’s twice, so there’s only a 1/20*1/20*1/20 = 1/8000 chance of getting a third 1. But that calculation explicitly makes the die rolls dependent. In essence, Pete is arguing that the previous rolls represent a-priori information that can be used to modify the probability of the next roll. Pete is a closet Bayesian in a Frequentist world.

But forget boring statistics jabber – look at the superstition on its own terms. Pete rolled 1,000 dice, not just one, and so if you roll each one three times you would have a total of 3000 rolls, out of which 3000/20 = 150 should be 1s. Note that Pete set aside 50 dice that rolled a 1 after the first round, so there are 100 1s still unused. Pete rolls only those 50 dice again, and gets about 3 that roll 1s, so now there are still 97 1s left! Of course you could argue that Pete has only made 1050 rolls thus far, in which case there are only 53 1s expected, but if you make that argument then you’ve admitted that each roll is independent and thus the next roll would still be a 1/20 chance of a 1 again. Plus, you made all those 97 1s angry by not carrying out the other 1950 rolls. Don’t anger the dice, Pete.