Biased Dice

[Christian]

100 rolls do not come even near to the statistically necessary amount of rolls to determine whether a d20 is biased or not.

Well, yes, I know that. I only did 100 on the first die; it may actually be slightly biased, but either to a small degree or to a larger degree that happened to not show up in that small number of rolls. That's why, when the first 100 rolls indicated a bias in my black die, I started some more sets ... I'm not sure the die I rolled only 100 times isn't biased, but I'm quite sure the other one is.

 

[devilish]

How do you do a chi-square test in Excel? Can't seem to find
a how-to anywhere?

Also, how many times would you need to roll it for accuracy sake?

Thanks

 

[GuardianLurker]

I believe chi-square is one of the built in functions. If it isn't, not-to-fret, for gaming purposes an eyeball-check on deviation from the expected flat frequency curve generally sufficies.

In theory, thousands. In practice, more than you can probably stand. I've found that you need a *minimum* of 10*N for a dN. It WON'T be accurate, but it will pop out dice you probably need to worry about. A reasonable (but still inaccurate) test would use 20-40*N. The more-trouble-than-its-worth test would use 100*N.

Of course, if you use a program to generate the numbers, crank the numbers as high as you want - the computer rolls dice a *lot* faster than you do.

 

[joshwitz]

Our DM runs a 'you roll the monster's attacks against you' mechanic. I'd been using this pretty blue d20 with white lettering, and one day I felt like it was persecuting me, rolling too many 1's. In a fit of pique after a particularly bad fumble, I told it "That's it! From now on, I'm using you only for my enemy's attack rolls!" to the great amusement of my party and my DM.

Ever since then, its been noteworthy when it rolls a '1', which I must say happens frequently: my character is the one in full plate up at the front of the party, drawing as many attacks as I can so that the bad guys aren't swinging at my other party members. Especially against 'fodder-level' creatures, who only hit me on a 20, but give up AoO's on a '1' using our fumble system. My DM's even started making cracks about banning 'that blue die'.

I can't say it rolls a '1' more than 1 time in 20, because I've never run it through Excel as you just did. But... it does sometimes feel like it.

I HATE that die. It's saved you on more than just a few occasions. I seem to recall that you sometimes lend it to other players (Darmaon?) too when they are in dire straits. Especially since a '1' in our game is a fumble AND we use a fumble table. The bad guys just seem to fall all over themselves when you pull that one out.

The Ratmen of Scarn curse the day the die of clumsiness was minted...

In another game, I once saw a player, in a pique of rage, toss a die out a 3 story window after a particularly bad series of rolls. We all game him this look ->

 

[Christian]

1) Different versions of Excel have different statistical analysis packages. It's a useful tool even if it doesn't have any statistical functions in your version, because chi-squares are calcluation intensive operations, which is something spreadsheets are well-equipped to deal with ... To calculate a chi-square of a distribution, you just add up the squares of the differences between the expected number of rolls of each number and the actual number of rolls of that number, divided by the total number of rolls. Eg. for the two-hundred roll stage of my test (when the expected value is ten rolls of each number on the d20), I had numbers that looked like this:

Value Rolls Square of Difference
1 11 1
2 10 0
3 15 25
...
18 16 36
19 10 0
20 9 1
___________________
TOTAL 200 4200
Chi-square value: 4200/200 = 21.0

Then, you can look that up in a chi-square table in your handy-dandy statistics textbook & see that for nineteen degrees of freedom, this is in the 30-35% range of randomness. (You don't have a statistics textbook handy? Cretin!) => There is a significant non-random component to the results from this die.

Actually, my version of Excel includes some chi-square statistical functions, that make life a bit easier. CHITEST(results table, expectations table) gives you the randomness percentage from a table of actual results and a matching table of expected results ... With dice, you're hoping that the CHITEST() output is very close to 1.

2) How many rolls you need depends on how small a bias you want to be able to detect, and how certain you want to be of your conclusion, as with any statistical test. Rule of thumb with chi-square is, at an absolute minimum, an expected value of 5 or more for every possible value in your raw data. This means 100 rolls for a d20. In practical terms, this means you start with 100 rolls, and continue in 100 roll batches until you feel happy. With my first die, my first 100 rolls were in the 95%+ random region. That left me feeling confident that the die was probably fair, or that any bias was at least pretty small. My first 100 rolls with the black die have me a randomness percentage of less than 50%, which inspired me to continue doing tests; each subsequent 100 roll set, while sometimes showing a higher randomness percentage individually, resulted in an increase in the overall total chi-square; this made it rapidly clear that the initially perceived non-randomness was real and not a statistical artifact. 500 - 1000 rolls should be plenty for anybody, unless you're hoping to open a casino using those dice ... A bias small enough to pass through that level of testing probably is not significantly impacting your gaming.

 

[joshwitz]

Coincidentally, I was in the Pirate store in San Francisco yesterday, and found a set of loaded dice! They always roll '1's. I can't wait to use this in my game next month. (but only as a joke!)

 

[TheAuldGrump]

I think that my dice are situationally biased. When I am rolling for the goblins lying in ambush I get threat followed by critical time after time... But for the climactic battle with the big villain? He prepares for the fight by standing on a bannana peel and buttering his sword. I don't know how many times it has happened, but my players have almost come to count on it.

The Auld Grump, don't ask what happened to the Sword Mage in Birthright... just don't ask.