Statistics is sometimes described as inverse probability.

In a typical probability problem, one starts by positing that a certain quantity has some given probability distribution, say the number of people entering a bank branch follows a Poisson distribution, and then goes on to compute probabilities such as the chance that more than 100 people (max capacity) require service at the same time. In a typical statistical problem, one observes the distribution, that is to say, the number of patrons over a period of time, and then finds a model to best represent the observed pattern.

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In Charles Wheelan's Naked Statistics (I'll post a proper review when I finish the book), he offered readers the following:

If you flip a fair coin 1,000,000 times and get 1,000,000 heads in a row, the probability of getting a tail on the next flip is still 1/2. (p. 102)

I'm here to tell you why it isn't 1/2. And it has to do with the difference between how a probabilist thinks and how a statistician thinks.

A probabilist starts with a couple of truths:

- the coin is fair
- each coin toss is an independent event

and from there, he or she computes the probability of different outcomes: one such outcome is to obtain 1 million heads followed by 1 tail in 1 million and one tosses. The probabilist tells us that that outcome is extremely rare but possible.

Now, if I find that 1,000,000 coin tosses produced 1,000,000 heads, I reject the notion that the coin is fair! Based on the observed data, I am more comfortable believing that the coin is severely biased towards heads. Therefore, my expectation of the next throw would be a very high chance of heads -- I'd certainly not conclude that the chance of heads is 50%, as Wheelan said there.

What Wheelan said is probably very commonly taught in statistics classes. This is unfortunate because in statistics, we start with the data, and figure out which probability distributions would be most consistent with the data. This is the inverse of probability modeling, in which one starts with the probability distribution.

It would be doubly unfortunate if that kind of statement shows up in a Bayesian textbook but I suspect you can find examples of that too in the section on probability.

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If you want to understand how statisticans think, this is a great place to start.

It absolutely is 1/2 by definition of a "fair coin". It's just staggeringly unlikely that you would ever get to this point in reality. By all means reject the premise that the coin is fair, but then you're not objecting to the idealised statement in the book but to a different one that you've created.

Posted by: Jamie Bull | 04/22/2013 at 09:25 AM

Spoken like a true probabilist. It's unfortunate that probabilists like to use euphemisms like "coin" for things that they *define*. It confuses the rest of us who think that a "coin" might be some actual physical object.

Posted by: Daniel Lakeland | 04/22/2013 at 08:21 PM

Something I read recently pointed out that a consequence of a university probability education was believing that after a coin toss came up 100 heads in a row, that the probability of heads on the next toss was 0.5, when anyone with real experience of gambling would conclude that there was something wrong with the coin.

Posted by: Ken | 04/23/2013 at 05:21 AM

It seems to me that in your explanation above, the statistician and the probabilist are working the problem from opposite ends. The statistician is starting with the sampled data and inferring something about the population and sampling frame (in this case, the coin and the sampling procedure). The probabilist is starting with known characteristics of the population (i.e. the coin is fair) and the sampling procedure (each sample comes from an independent event), then estimates the probability of seeing a particular sample distribution. Is this fair, or have I misunderstood something?

Posted by: Tom | 04/23/2013 at 11:14 AM

Tom: Exactly. That's the origin of the term "inverse probability".

Posted by: Kaiser | 04/23/2013 at 11:17 AM

What if the fairness of the coin was properly established with some robust scientific method and then it came up heads 1,000,000 times? Extremely unlikely, but we are talking probability concepts here, so it is conceptually possible. I guess the book's author was trying to demonstrate that even if a guaranteed fair coin had a long streak, that does not influence the next single toss outcome.

In regards to betting on that single outcome, I'd say people who rush to make bets in such cases are betting on the low probability of the streak continuing, rather than on the constant 0.5 of the next toss.

Posted by: Dimitri | 04/23/2013 at 09:00 PM

Good explanation of the difference between probability and statistics. I wish I had this many years ago. I am only now getting a grasp on the difference between probability and null hypothesis significance testing (NHST) versus exploratory data analysis and Bayesian thinking. I was taught the former, when what I often want is the latter.

Posted by: Jordan | 04/24/2013 at 11:14 AM

The comment by Dimitri above is interesting because it reveals a form of innumeracy. (1/2) to the millionth power is not "extremely unlikely," it's in any real physical sense impossible.

I will throw in one more twist. We are given three incompatible statements:

1. The coin is "fair." This concept is not clearly defined, but I will take it to mean that, when flipped, there is a 1/2 chance it lands heads.

2. It was flipped 1 million times.

3. All flips turned out heads.

Any two of these three statements can be true, but not all three.

One possibility is Kaiser's, that 2 and 3 are true, hence 1 is false. Another possibility is that 1 and 2 are true, but 3 are false. After all, how would we know that a coin came up heads a million times. We're taking someone's word for it. See Section 3 of this paper for a similar example.

Posted by: Andrew Gelman | 04/28/2013 at 08:14 PM

Andrew: I like your formulation.

Posted by: Kaiser | 05/01/2013 at 12:20 AM

In my daily experiences with data (albeit not a large sample size) I don't know if the coin is fair. To extend the analogy, I'm usually flipping some foregin bottle cap with little prior knowledge as to whether the cap comes up on the "Presidente" side more often than the other side.

Posted by: Jordan | 05/01/2013 at 02:54 PM