I'm back with more stats on Travis' rolls- it's the Eleventh Hour this time!

Oh man, I get to talk about Bayesian statistics!

So Bayes' theorem states that P(B|A) = P(B)P(A|B)/P(A). Let's let A be "Magnus getting rolls this good" and B be "Travis is cheating". Note that P(A) = P(B)P(A|B) + P(~B)P(A|~B) (where P(A|B) is "the probability of A happening given that B happened", and P(~B) is "the probability of B *not happening". Therefore, P(B|A) = (P(B)P(A|B))/(P(B)P(A|B) + P(~B)*P(A|~B))); in other words, the probability that Travis is cheating given that his rolls are this good is equal to the probability that he would be cheating at all multiplied by the probability that he'd get rolls this good if he were cheating, all over the denominator (the probability product I just mentioned) plus the probability that he's not cheating multiplied by the probability that he'd get rolls this good if he wasn't cheating.

If none of that made sense, that's because I'm bad at explaining things.

There are a couple probabilities we need to find in order to figure out how likely it is that Travis is cheating. Let's start with the easy ones. P(A|~B) was just provided by b1ak3: 2.6*10-12. P(A|B) is also straightforward - it is safe to assume that were Travis cheating, he would want to do well. However, there may be some insinuating circumstances that prevent him from getting rolls this good; perhaps he wants to be subtle, or he doesn't want to cheat very often. I'll make it slightly easier on Travis, and say that P(A|B) = 0.8 (80%).

There's one last thing, though, and this one's the doozy. We need to find the odds that Travis would be cheating, pretending we know nothing about his rolls. This is the source of the controversy here - different people find it more or less believable that Travis would even cheat in the first place. Rather than simply state a value for P(B), I'll do this: I'll find the value for P(B) for which P(B|A) would equal 0.5; in other words, I'll find the threshhold for trust of Travis; your P(B) must be lower than this number to believe that Travis is not cheating.

Now we can do some calculations. Keep in mind that P(~B) = 1-P(B), so

0.5 = (P(B)0.8)/(P(B)0.8+(1-P(B))2.610-12) (0.4-1.310-12)P(B)+1.310-12 = 0.8P(B) P(B)=(1.310-12)/(1.310-12+0.4) P(B)~3.25*10-12

In other words, if the likelihood that Travis would ever cheat is greater than 0.000000000325%, then he's probably cheating.

Of course, this is completely reasonable, and probably quite benign. I don't quite get the argument that there's a weighting issue - his dice make way too much noise hitting the table for them to not be metal, though I suppose I'd have to use Bayes' theorem again to figure out exactly how unlikely that is.

Truth be told, I don't actually care whether or not he's cheating, I just like talking about Bayesian statistics.

In other news, all of my friends have stopped taking to me recently.

/r/TheAdventureZone Thread Parent