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.

5 thoughts on “pre-rolling the dice”

  1. ah I hadnt seen it formulated that way before. yeah in a sense D&D is a gambler’ game, albeit one that favors the gambler.

    that wiki entry you linked notes that there are some things that try ot trick you into thinking the Gambler’s Fallacy applies when in fact it does not. An example of this is the Monty Haul problem.

  2. I remember seeing one of those shows where random morons try to cheat Vegas, and someone pointed out that there was one group of people that figured out a pattern on a particular roulette wheel. They had spent months gathering a statistically significant sample of this particular table’s results, and eventually won lots of money. The idea was that imperfections in the table’s design (i.e. it’s not perfectly level, the wear and tear is not uniform, etc…) could favor specific results.

    So theoretically, imperfections in the manufacturing of a particular 20 sided die could favor a specific number (or avoid rolling a 1, etc…). Of course, statistically speaking, you’d have to have a large sample set, and if you had 1000 dice, you’d have to do millions and millions of rolls before you figured out which ones didn’t roll a 1 very often.

    The Gambler’s Fallacy is funny though, especially at the blackjack table. Good players know the basic strategy by heart and so when someone doesn’t play by basic strategy, they can still see what the next cards that come out are and will often blame the person not playing by basic strategy for losses. However, a person hitting or staying has just as much a chance of helping as hurting… I guess people just notice the bad results better.

  3. Nah, Pete is doing Bayesian wrong.

    A Bayesian would take the opposite approach – roll the dice, and keep the one that rolls twenty twice in a row. Because no die is perfect, a die that rolled two twenties is has a slightly higher probability of being favourably loaded, and so his posterior probability of rolling twenties with the ‘twenty-rollers’ is a bit better than with all his other dice.

  4. I’m sure I’m a few years late with this, but I stumbled across this somehow and figured I’d add my own thoughts on this.

    Even if Pete were somehow right in that the three roles are somehow connected (they aren’t), what he fails to realize is that while rolling three ones is a one in eight thousand chance, rolling two ones and a two is also one in eight thousand. Or two ones and a three, or two and a four, etc. No matter what the third roll comes up as, the results of any three rolls are always one in eight thousand, making one in eight thousand odds not as rare as you’d think.

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