The lottery fallacy.

Understanding common logical fallacies is really helpful to quickly put your finger on the exact flaw in a fishy argument. But sometimes fallacies themselves can trick you, and you can end up committing the fallacy fallacy by accusing someone of a fallacy who isn't actually guilty of it. Let's take the lottery fallacy, here is a simple example of it from Lucid Philosophy:

I conclude that Bob must have cheated when he won the lottery because the odds of him winning were twenty million to one.

Indeed, there are millions and millions of lottery tickets bought, and it's not at all uncommon for there to be a winner, even though the chance for any particular person, like our Bob, to win is incredibly small. So just because some person won against incredible odds doesn't give us the right to infer that the lottery was somehow unfair, rigged. We don't need some special explanation for the incredibly unlikely event of that specific person winning, someone was likely to win after all! Lucid Philosophy puts it lucidly this way:

The lottery fallacy arises when we invalidly infer x must be designed because x is improbable.

Another useful formulation a commenter on another blog proposed is: just because some event is wildly improbable doesn’t mean it is significant if it is drawn from a huge population of similar effectively interchangeable events. While according to the "official" formulation above multiple trials are not an essential component of the lottery fallacy, most cases of it do involve multiple trials.

So far pretty simple right? If you are yawning now it's time to wake up, because we will now look at a couple of cases where many people start to get confused. We will use them to illustrate an important idea that goes far beyond the lottery fallacy: the idea that sometimes imprecise language can lull you into a false sense of understanding.

Case 1. The lottery wife.

Suppose the person who won a hundred million to one lottery is the wife of the person in charge of conducting the lottery. Can you conclude the lottery wasn't conducted fairly? Well, you can't be 100% sure it was rigged but you would probably increase your degree of belief (aka credence) that perhaps some shenanigans were involved. 

Mr. Smartypants: 

But you are committing the lottery fallacy! Someone had to win, and the organizer's wife was as likely to win as some random Joe Smith. If you wouldn't suspect shenanigans in the case of Joe, you can't treat the wife with a double standard. Don't you know your fallacies?

Is he right, are you unfairly discriminating against the poor lottery wife? Actually no. It would be a lottery fallacy to suspect rigging because Joe Smith won, but not if the winner is the lottery wife. What is the difference, aren't they equally likely to win?

The answer to this question is key to fully understanding the lottery fallacy: 

1. They are equally likely to win if the lottery was fair.
2. They are not equally likely to win if the lottery was rigged.

You might think there is a simpler way to say this: Joe Smith is not special as far as the lottery is concerned, but the wife of the person conducting the lottery certainly is. While that's not wrong in this case, we will soon see why understanding this fallacy in "fuzzy" terms like "special" leads people to make big mistakes in other situations.

Case 2. A bajillion dice.

Let's get slightly more advanced. Suppose you walked into a room and you saw 50 dice scattered on the table. You believe that your messy roommate, who always buys weird items, threw them on the table randomly. Let's call the hypothesis that the dice were tossed randomly R. Suppose then you come closer and realize that all the dice are showing a six - would you then decrease your credence in R? In non-geeky terms, would you now be less inclined to believe in truly random tosses? I assume you would - after all, another possibility is that he or someone else deliberately arranged them to show all sixes. Or, who knows, maybe something even weirder happened. Whatever could have happened, if it's not R, let's just call it NR (not random).

By the way, why use all these terms, like hypothesis or credence, why the letters like R and NR? Why don't I just say: you would conclude that someone arranged the dice? It's not to make simple things seem confusing, I promise. It's because this language is the key to avoiding subtle mistakes that trip up many people, even with very good critical thinking skills. A good example of this is the recent disagreement between philosopher Philip Goff and Dr. Steven Novella, which has lasted unresolved for weeks and generated thousands of comments on Steven's blog. 

So let's collect all the pieces and express them compactly. We have two hypotheses and a piece of data:

• R = random dice tosses
• NR = not R
• D = 6,6,6,6,... appearing on fifty dice

And we have a question, which can be expressed in a few ways:

• Does the data D lend support to NR?
• Is D evidence for NR?
• Should learning D increase our credence for NR?

Intuitively it probably seems like the answer is clearly yes, since the probability that all 50 dice would randomly land all sixes is ludicrously low by chance alone, since there are 6*6*6*6... = 6^50 possible ways they could have landed, an astronomically large number. But Mr. Smartypants is ready with another accusation:

It's the lottery fallacy! Would you think any shenanigans had to be involved if the numbers were 3,4,6,6,2,1,6,4,2,...? Some numbers had to come up, and 6,6,6,6,6,6,6,... only seems very surprising because you arbitrarily assign some significance to it. But objectively speaking this sequence is exacty as likely as 3,4,6,6,2,1,6,4,2,... 

While most everybody would agree that no lottery fallacy arises in the lottery wife case, the dice scenario seems to put people in two opposite camps. That's exactly what happened in recent discussions on the Neurologica blog, with many people expressing Mr. Smartypants' exact argument. But after the lottery wife case you might know how to respond to that argument:

1. Sure, those two sequences are equally likely if the tosses are random (R).
2. But our D (6,6,6,6,...) is far more likely than the average number if the tosses are not random (NR).

Mr. Smartypants' mistake is only focusing on 1 and erroneously believing that 1 means the lottery fallacy is committed. I hope the lottery wife and the dice cases make it clear 1 means no such thing, everything stands or falls depending on 2. 

The perils of fuzzy language.

But now try your hand at a more sophisticated objection someone offered, which I will paraphrase:

In the lottery wife case there was something objectively special about her compared to some random Joe. She is the wife of the guy who runs the lottery, that's pretty special! But there is nothing objectively special about D, the only reason you are surprised to see 6,6,6,6,... is because you subjectively attach more significance to this number.

The language about subjectivity, surprise, and specialness is very common in these discussions, even by professional philosophers. In simple cases it does the job, but in more subtle cases the imprecise language leads to mistakes and to people talking past each other. To address the above objection, we need to draw clear distinctions between three different meanings of "special", which the objection above conflates:

1. R-special = D is actually not that improbable if R. 
2. NR-special = D is not that improbable if NR. 
3. Subjectively special = D is more recognizable to me, more significant. 

Example of R-special. For the dice case that would say: if the tosses were random 6,6,6,6 is somehow still more likely to win than the average sequence. In both of our two stories this is not the case (by assumption, otherwise there wouldn't be much to explain).

Example of NR-special. If the lottery is rigged the wife is not astronomically improbable to be the beneficiary, being closely related to the likely rigger.

Example of subjectively special. Suppose the only one who could have arranged the dice is a child who doesn't know you, and suppose you found the dice in a sequence that would look completely random to the child but had some personal significance to you. In this case the sequence would be subjectively special to you but not special in the other two senses.

It's this last case that describes the lottery fallacy. In that case the presence of the only possible rigger, the child, doesn't help at all to explain the sequence! Because the sequence would look completely random to the child he would be no more likely to produce it than random chance would. Therefore it would be fallacious to use the subjective specialness of the sequence as evidence that the child rigged it. We can now summarize the distinction between the lottery fallacy and a valid inference very compactly:

Valid inference. If D is NR-special, then D is evidence for NR.

Lottery fallacy. If D is not NR-special, but only subjectively special, inferring NR is illegitimate.

With these senses of special neatly separated, the above objection reduces to this:

D (6,6,6,6,...) is subjectively special but not NR-special.

Now this is easy to answer: if this were true then, indeed, inferring NR would commit the lottery fallacy, but this simply is not true: in any reasonable context 6,6,6,6,6,... on fifty dice is NR-special. Recall what NR-special means. It means that if the dice weren't just tossed randomly (i.e. if NR, so roughly speaking if somebody arranged the dice) then the specific configuration (6,6,6,6,6...) wasn't that unlikely to occur, at least compared to how astronomically unlikely it were to occur in the case of random tosses. In virtually any context we expect somebody who would arrange the dice to be significantly more likely than "dumb luck" to arrange them in all sixes (or any other specific "nice" pattern). 

Conclusion.

I hope this has been helpful in giving you a clear and precise understanding of the lottery fallacy, and perhaps in clarifying some more general concepts such as evidence and credence. You might wonder: where would this apply in real life? Are there any serious questions where reasonable people could make a mistake either by committing the lottery fallacy or by erroneously believing someone else is committing it?

There definitely are. A recent article about whether fine-tuning is evidence of the multiverse by Steven Novella has inspired a couple of other articles on this blog, because this topic involves a host of very fundamental issues about evaluating evidence under very tricky conditions. So tricky in fact that good critical thinkers continue to have wild disagreements even in very simple and exactly specified scenarios that don't require any knowledge of physics or advanced math. The lottery fallacy plays a central role in these disagreements. If you want a really tough challenge to practice your understanding of the lottery fallacy, I invite you to check out Steven's article, as well as an article by his opponent in the debate, philosopher Philip Goff, and try to figure out who is right.