How Probability Works

From Barry Greenstein’s insightful poker book, Ace on the River:

Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and all ten times it comes up heads. What is the probability that it will come up heads on the eleventh flip?

A novice gambler would tell you, “Tails is more likely than heads, since things have to even out and tails is due to come up.”

A math student would tell you, “We can’t predict the future from the past. The odds are still even.”

A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it, but I’d bet someone else that heads will come up again.”

Yes I know the math student would really say “individual trials are uncorrelated,” not “we can’t predict the future from the past.” The lesson still holds.

Happy Labor Day, everyone.

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35 Responses to How Probability Works

  1. Spencer says:

    On Thursday I composed a musical soundscape called “Cosmic Variance.” Now what are the odds?

  2. Fred says:

    A quantum physicist would say, “100%, it is guaranteed to come up heads in one universe, and tails in another.”

  3. Fred says:

    An inflationary cosmologist would say, “It is impossible to compute probabilities in the eternally inflating multiverse and so I can’t give you a proper answer. If I do give you an answer, rest assured it will lead to non-sensical paradoxes, which will allow me to write many new follow up papers.”

    Sean, speaking of such things, what are your thoughts on 1108.3080?

  4. Tom of the Sweetwater Sea says:

    The coin has no memory. It does not know what it did last time.

  5. STAN says:


  6. Fred says:

    Actually Sean, a math student probably would say “We can’t predict the future from the past.” While a statistician, or a sensible physicist (such as an experimentalist) would be the ones saying “individual trials are uncorrelated.” Mathematicians are smart, but are notoriously bad at understanding how things work in the real world. They hate making approximations etc. Thats why they didn’t pursue science, but studied pure mathematics, and wouldn’t dare do something as imprecise as making a real world prediction.

  7. HappyEvilSlosh says:

    A mathematician would say “It yet hasn’t been established the coin is fair”, where you proceed from there depends on whether you subscribe to the frequentist or bayesian school of thought. Uncorrelated trials generally make it easier, not more difficult, to calculate probabilities.
    (Also not all mathematicians are pure mathematicians)

    –a mathematician

  8. Stu Savory says:

    Just FYI : a question I ask my students is
    “Given a single biased coin, how can you make fair tosses?”

    Fair meaning 50:50 outcomes within the bounds of probability theory.

  9. Fred says:

    A creationist would say, “Probability is only a theory. God decides which side the coin comes up.”

  10. socle says:

    I understand Greenstein’s point, but this mathematician would have agreed with the professional gambler, even as a student. I would think that pretty much any competent person would, tbh.

  11. puzzled says:

    Stu: Make a pair of coin tosses. If the outcome is TT or HH, discard the pair and try again. If the outcome is TH, pronounce this to be tails; if it is HT, call this heads.

  12. Cusp says:

    While each flip is independent, a Bayesian would use one to update their knowledge of the fairness of the coin.

    Knowing the past does eventually make a prediction about the future.

  13. It depends on the prior information. I’d say in most situations the correct answer is the third one.

  14. Jorge says:

    Bernanke would say: “Let’s just make more coins and see what happens”

  15. Tatek says:

    I agree with Jorge. I would extend the argument and propose that we create a bigger world and then see what happens

  16. Roland Kie says:

    A determinist would say ‘It is calculable what it would turn out to be according to how the person’s movement, the movement of the surrounding air, the weight and side of the coin before it was flipped, the moment when the person catches the coin etcetc’ but, of course since he won’t heve the power to calculate he wopn’t really have an answer and…then it goes nowhere…slow.

  17. jt512 says:

    A statistician wouldn’t use the word “uncorrelated” when the question is about independence. Hopefully, a mathematician who had taken a mathematical statistics course wouldn’t either.

  18. Fred says:

    A statistician would use the word “uncorrelated” knowing that it obviously means the same as “independence” when the random variable is binary: heads or tails (although jt512 does not seem to know this trivial fact, even though jt512 claims to be such an expert on statistics and is so critical of other people’s comments).

  19. BS vs ML says:

    What is really interesting is two different inductive processes competing.


    Black Swan vs Moore’s Law

  20. A theoretical computer scientist might say: “choosing a prior distribution over possible sequences of coin flips where each sequence gets weighted by 2 to the minus its Kolmogorov complexity, and given the outcomes of the first ten flips, we find that ‘heads every time’ gets assigned an overwhelmingly large posterior probability; therefore, I agree with the professional gambler.”

    General Lesson: Any time we find that “math” disagrees with reality, the problem is never with “math”—it’s with us, for using the wrong math! 🙂

  21. Tony P says:

    We can do this – let’s check the math folks – a Bayesian would find the probability of heads to be 11/12?

  22. Ken says:

    Don’t forget the gamer’s explanation. It combines the novice gambler’s naive understanding of probability with the profession gambler’s immediate “I can exploit this to my advantage”.

  23. jt512 says:

    Actually, the question isn’t about correlation or independence; it’s about what the probability of the next flip being heads is. However, if the probability of “heads” is 1 (as it appears that it may be), then the trials are still statistically independent, but their correlation is undefined. So, no, Fred, “uncorrelated” does not mean the same thing as “independent” if the trials are Bernoulli.

  24. jt512 says:

    @Tony. I concur: 11/12, using a uniform prior.

  25. Austin Frisch says:

    As a former professional gambler, I’ve had the best of it after going all and lost 10 times in a row, on more than one occasion. Typically, I then proceeded to dig out some of my teeth with a grapefruit spoon.