Uncertain Urns Puzzle

A familiar probability example, using urns, is adapted to illustrate ‘true’ (non-numeric) uncertainty.

Simple situation

The following is a good teaching example:

Suppose that an urn is known to contain black and white balls that are otherwise identical. A subject claims to be able to predict the colour of a ball that they draw ‘at random’.

They ‘predict’ and draw a black ball. What are the odds that they are really able to predict?

From a Bayesian perspective, the final odds are the initial odds times the likelihood ratio. If there are b black and w balls and we represent the evidence by E and likelihoods by P( E | ), then P( E | Predict ) = 1 and P( E | Luck) = b/(b+w). Thus the rarer the phenomenon predicted, the more a correct prediction tends to support the claim, of reliable prediction. 

Common quibbles

There is, however, some subjectivity in the estimated probability that the subject can predict:

  • In this case, the initial odds seem somewhat arbitrary, and Bayes’ rule seems not to apply. For example, have you considered that the different colours may result in different temperatures? Such a thought is not ‘evidence’ in the sense of Bayes’ rule, but might change your subjective estimate of the probability prior to their draw.
  • If we do not know the proportions of black and white balls for sure then the likelihood is uncertain.

 Multiple urns

Here we introduce a different type of uncertainty:

Suppose now that the subject is faced with two urns and selects a ball from one. Given the number of black and white balls in each urn, what is the likelihood, P( E | Luck ), of a correct prediction due to luck?

If you think the question is ambiguous, please disambiguate it however you wish.

Suppose you know the total numbers of black and white balls in the two urns. Is the likelihood estimate P( E | Luck) = b/(b+w) reasonable? Could it be biased? How?

See Also

A legal example  Other, similar, puzzles.

Dave Marsay 


About Dave Marsay
Mathematician with an interest in 'good' reasoning.

5 Responses to Uncertain Urns Puzzle

  1. A problem covered by Andrew Lo in this excellent presentation:

    “Physics Envy…” presentation by Prof Andrew Lo: Risk & Uncertainty (in financial systems) http://wp.me/p16h8c-j


    • Dave Marsay says:

      David, I think Lo’s presentation tremendously important, and I may blog on it. But I have a general difficulty and a specific one. The general difficulty is that he consigns mathematics to the lower levels of uncertainty, whereas I think Keynes’ Treatise is useful more broadly. What else is there? My specific difficulty is that in his urn example he seems to insist on certain assumptions that seem dubious to me. I agree that my example is related to his, but I hope that in my case it is more obvious that Keynes’ objections to his principle of indifference are valid, and that the principle gives the ‘wrong’ answer.


  2. Blue Aurora says:

    Ah, the fabled urn problem…incidentally Dr. Marsay, have you read Daniel Ellsberg’s doctoral dissertation?


  3. Dave Marsay says:

    No. Thanks for the link. I have just read and commented on his 1961 paper, at https://djmarsay.wordpress.com/bibliography/rationality-and-uncertainty/more-rationality/ellsbergs-risk/ . Ellsberg seems to have intended his urn examples to show that it was reasonable to take account of ‘true’ uncertainty, not just probability. But his arguments are not mathematical and seem to have failed to convince, e.g. Wikipedia. My point is different. I would like to think that it would be better understood.

    Does his thesis have any mathematical arguments or discuss Keynes’ work in this area?

  4. Blue Aurora says:

    Ellsberg’s doctoral dissertation was in Economics, but yes, it does use mathematical arguments. There’s a “restricted Bayes-Hurwicz criterion” in his doctoral dissertation and he pokes a hole in the Savage axioms. And yes, he does cite Keynes’s Treatise on Probability in his doctoral dissertation.

    But did you read Ellsberg’s 1961 article, or no? If not, then it’s here.


    I recommend reading the dissertation, as it is of course, far more developed than his article in the Quarterly Journal of Economics. Ellsberg’s arguments have generated a lot of research, if I’m not wrong…

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