Statistics vs. Probability – Did POTRIPPER Cheat?
This is a bit pointless, as I’m blogging about a “controversy” that has already been settled. I do think there are useful tools to take from this incident, though.
For those out of the loop, a few days ago players of online poker site “Absolute Poker” (no link, deal with it) accused a player called “POTRIPPER” of playing while being able to see the hole cards. I’ll save the poker laymen’s task of trying to figure out what that means (as I had to) – in that variant of poker some of the cards are shown, while others are hidden. People were accusing POTRIPPER that (s)he was playing while being able to see everyone’s hidden cards.
I should point out that this is a settled controversy. Absolute Poker admitted that this is the work of an internal security breach. I am less interested in the specific case, however, than I am in looking at tools designed to answer the “how can we know” question.
For those of you who are not up to speed, here is the slashdot story, and more importantly, here is an analysis by a probability expert, as well as a skeptic’s view.
Lets start with our skeptic. Justin West’s point, in a nutshell, is that with enough people playing enough games, the variations between good games and bad games is bound to create, through no more than the fluke of randomness, and certainly through no cheating, a situation where someone gets this lucky and wins, consistently, for several games in a row. He also has a bunch of other claims, which are not probability related, but I think they are irrelevant. Again, just to get them off the board, he says that Absolute Poker has no interest in allowing such cheating. It turns out he was right about that point, yet wrong about the conclusion. So it goes.
Back to the statistics, however, and to The Wizard of Odds, aka Michael Shackleford. Michael analyses the results and comes up with the following conclusions. The chances you will play a series of games with the results POTRIPPER got are lower than the chances of you buying six lottery tickets (participating in six different drafts) and striking out on all six. That is pure probability calculations, and is (I’m assuming) backed by hard science (well, math at least).
So, does that mean POTRIPPER cheated? Answer – we don’t know.
Here’s the thing. In our universe it is entirely possible that both Michael and Justin are simultaneously right. Low probability events, even extremely low probability events, may still be extremely likely to happen. To loosely quote someone (I believe it was Greg Egan, but I’m not sure), if you take an event that has a 1:1,000,000 chance of happening over the course of a day, over the past 24 hours 6 thousand people experienced it (give or take a few). The event is extremely unlikely, but there are enough people to make sure it will happen a lot. This is what Justin claims has happened here.
Of course, now that Absolute Poker has come clean, we know Justin was wrong. I claim, however, that we could have known that even before they did.
In order to understand why, let’s understand what probability is. Probability is a branch of mathematics that attempts to answer the question “what are the chances that X will happen”. Probability handles mostly full-knowledge cases. What probability is not equipped to do is to handle real life cases. It can create a model with which you can compare, but it cannot analyze real life results. For that, you need statistics.
Unfortunately, statistics is not as clear cut and precise a science as probability is. It does provide us with some tools, however.
Let’s go back to Justin’s claims. Justin says that with enough people playing, a single player may get that luck over the course of two days by pure chance. In order to asses whether that is true, we really should know several bits of information we really don’t. We need to know how many people are playing Poker, how long, etc.
And this is where it really gets murky. I’m not a gambler, so I don’t know the answer to some of the questions. However, suppose there is another game where players play against each other. Is there any reason to assume that, should someone got lucky there, the same response wouldn’t come? In other words, we cannot limit our search to poker players alone, as any game that allows this type of cheating would create this kind of outcry should someone get lucky.
Exactly how lucky did POTRIPPER supposedly get? Michael does give us an answer to that one. Luckier than someone who buys exactly six lottery tickets, and wins all six. That’s pretty damn lucky. Still, with enough people, that should happen by accident, shouldn’t it?
Michael assigns a probability of 1:1.8×1044. I’m no probability expert, so I will accept that number at face value. What does it mean, however? Is it possible that Justin’s assertion, that it’s possible to get that lucky by, well, luck, is the right answer?
Here’s my answer. About five years ago I read a small story in the paper about someone from Britain who won the lottery twice. I cannot, right now, remember whether he only bought two tickets or not, and that affects the probability, but it does give us a general idea. Since I’m trying to disprove, let’s take an assumption that will make it harder on me. Let’s assume that that person did, indeed, only buy two tickets in his life.
Since I don’t live in Britain, we can safely assume that had the person winning the Lottery twice not been from Britain, but from, say, Malaysia or Turkey, the news would still make it into the paper. I don’t remember the precisely how long ago I read that news, but it was definitely at least five years ago. I also remember about half a year ago, maybe slightly more, that I read such similar news again. Someone won the lottery twice.
What we learn is that:
- Someone winning the lottery twice is news worthy enough to make it into newspapers in other countries
- The frequency of someone, anywhere in the world, is about once every three years.
At that rate, it would probably take several thousands of years, probably bordering on the million, before we hear of someone winning the lottery three times. It is extremely doubtful we will hear of someone winning the lottery six times before the sun turns to a white dwarf, and certainly not with someone only buying six lottery tickets. Such an occurrence simply does not happen.
I’m going to assume one more assumption. I’m going to assume that more people buy lottery tickets (which are legal) than play poker online (which is illegal in many countries). As such, I think we can safely say that Justine’s reaction was somewhat understandable, but upon inspection, unfounded. Such an occurrence really CANNOT happen by chance, and I would be really surprised to see, as he claimed he can provide, plenty of other cases in which people got that lucky without some sort of foul play.