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Dodgson’s Paradox |
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In 1893 Charles Dodgson compiled a book of mathematical puzzles with solutions, which he entitled “Pillow Problems Worked Out During Wakeful Hours”. It consisted of 72 problems with solutions, which (he reports) he worked out in his head at night while waiting to fall asleep. While some people count sheep, he solved mathematical puzzles. He tells us that his motivation was not so much to aid in falling asleep, but to crowd out the vexing thoughts of the day. All the puzzles are elementary, although in the introduction he called special attention to the last problem in the book: |
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If any of my readers should feel inclined to reproach me with having worked too uniformly in the region of Common-place, and with never having ventured to wander out of the beaten tracks, I can proudly point to my one Problem in 'Transcendental Probabilities' - a subject in which, I believe, very little has yet been done by even the most enterprising of mathematical explorers. To the casual reader it may seem abnormal, and even paradoxical; but I would have such a reader ask himself, candidly, the question "is not Life itself a Paradox?" |
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With this intriguing introduction, we may be surprised to find that the Transcendental problem to which he refers is simply this: |
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A bag contains two marbles, as to which nothing is known except that each is either black or white. Ascertain their colors without taking them out of the bag. |
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This banal problem is obviously insoluble, and yet Dodgson confidently assures us that the answer is “One is black, and the other is white”. Well, this is interesting. How could this possibly be inferred from the given information? |
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Fortunately, he provides his reasoning at the back of the book. First, he asserts that the contents of the bag are either BB, BW, or WW, with probabilities1/4, 1/2, and 1/4 respectively. We could not actually infer this from the statement of the problem, but it’s at least plausible that, in the absence of any other information, we should assume the color of each marble would be chosen independently and at random, with equal probability of being black or white. On this basis, Dodgson’s probabilities follow. Now, he asks us to consider adding a black marble to the bag (without looking), so that the contents are BBB, BBW, or BWW with probabilities 1/4, 1/2, and 1/4 respectively. If, after adding a black marble, we draw out a marble at random, the probability of drawing a black marble is |
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So far so good. At this point Dodgson asserts (presumably tongue in cheek) that, since the probability of drawing a black marble is 2/3, the contents of the bag must be BBW (i.e., two black and one white), and since we just added a black marble, the original contents must have been BW (one black and one white). |
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To lend support to the paradoxical conclusion, Dodgson prefaced his explanation by asking the reader to agree that if the bag contains three marbles, each either black or white, and if the probability of drawing a black is 2/3, then there must be two black marbles and one white marble in the bag. Then, after showing that the probability of drawing a black is 2/3, he says it follows that there must be two black marbles and one white. |
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Dodgson’s example shows that the probability of a particular outcome (e.g., drawing black) of performing a certain operation on a system may depend on the state of the system, but it may also depend on our knowledge of the state of the system. It is possible that the actual state of the 3-marble system is (for example) BWW, and if we know this is the state of the system, then the probability of drawing black is 1/3. On the other hand, if we don’t know the state of the system, but we know that it is in state BBB, BBW, or BWW with probabilities 1/4, 1/2, and 1/4 respectively, then the overall probability of drawing black is 2/3. |
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The problem can be split into two stages, first randomly selecting the marbles (from some distribution) to place in the bag, and then randomly selecting one of the marbles from the bag. Once the first stage has been completed, we might think that the ambiguity in the outcome of the first stage has been eliminated, since we now have a definite set of marbles in the bag. At this point one could argue that there is an “actual” probability of randomly drawing black, based on the set of marbles that are actually in the bag, and this “actual” probability is 2/3 just if the contents of the bag are BBW. However, there is also a “contextual” probability of drawing black, from the standpoint of someone who doesn’t know what marbles are in the bag, so the contextual probability accounts for the different possible contents of the bag, given the known distribution for those different possibilities. Dodgson’s paradox is a play on these two different levels of probability. Indeed if we had some way of knowing that the “actual” probability of drawing black (after adding the black marble) was 2/3, this would be equivalent to knowing that the first two marbles were one black and one white. However, our reasoning has not revealed the “actual” probability, it has only told us that the contextual probability is 2/3, which is just the suitably weighted average probability over the different possible contents of the bag. The fact that the contextual probability in this case happens to equal the actual probability for one of the possible contents is just a numerical accident. |
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This sort of multi-level probability arises in many circumstances. For example, suppose there is a .05 chance that a faulty part was installed on your vehicle, and suppose that vehicles with the faulty part have a 0.01 chance of failing on the next trip, but vehicles without the faulty part have only a 0.0001 chance of failing on the next trip. What is the probability of your vehicle failing on the next trip? We might say the probability is either 0.01 or 0.0001, depending on whether or not the faulty part is installed. One of these is the “actual” probability – but we don’t know which one. On the other hand, the overall contextual probability is (0.05)(0.01) + (0.95)(0.0001), which equals 0.000595. |
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It’s tempting to think that Dodgson’s “transcendental probability” problem was just intended as a satirical comment on the debate between the Frequentist and the Bayesian approaches to probability theory. However, neither of these approaches has any difficulty resolving this paradox. |
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