Bag of Beans


The two sharply contrasting probabilities - 1/2 and 1/1,000,000 - for the million bean case suggest that this is not a genuine problem in probability, ie that there is no rational answer.

I believe that to be a valid probability problem, one that has a single correct answer that can be mathematically determined, a problem has to satisfy two conditions:

(a) Any randomisation process that is involved in the experiment is made explicit. (See Bertrand's paradox.)

(b) The experiment can, in principle, be repeated a large number of times. (This is the basis of probability.)

The bag of beans paradox does not satisfy either of these two conditions. Firstly, the method of randomising (or not randomising!) the contents of the bag is completely unknown. We cannot assume that the colour of each bean has been randomly chosen.

Secondly, the experiment cannot be repeated. If it could be repeated many times then getting a white bean every time a bean was picked would simply mean that all the beans were white.

If my two conditions are indeed necessary then there are many philosophical consequences. For instance, people ask what is the probability that life arose by chance on our planet. This question does not satisfy either condition and hence cannot have a mathematically valid answer.

Another example is the Anthropic Fallacy, the belief that the universe was designed in order that we human beings could have evolved within it. The argument is that there are a number of basic constants in physics, a small change in any of which would guarantee that life as we know it could never arise.

At the very least, the bag of beans problem suggests that probability is an extremely tricky beast, one that we must treat with maximum caution.

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