And Other Probability Paradoxes

Probability theory abounds in paradoxes that wrench common sense and trap the unwary. In this chapter we consider a startling new paradox involving the relation called transitivity and a group of paradoxes stemming from the careless application of what is called the principle of indifference.

Transitivity is a binary relation such that if it holds between A and B and between B and C, it must also hold between A and C. A common example is the relation "heavier than." If A is heavier than B and B is heavier than C, then A is heavier than C. The three sets of four dice shown "unfolded" in Figure 21 were designed by Bradley Efron, a statistician at Stanford University, to dramatize some recent discoveries about a general class of probability paradoxes that violate transitivity. With any of these sets of dice you can operate a betting game so contrary to intuition that experienced gamblers will find it almost impossible to comprehend even after they have completely analyzed it.

The four dice at the top of the illustration are numbered in the simplest way that provides the winner with the maximum advantage. Allow someone to pick any die from this set. You then select a die from the remaining three. Both dice are tossed and the person who gets the highest number wins. Surely, it seems, if your opponent is allowed the first choice of a die before each contest, the game must either be fair or favor your opponent. If at least two dice have equal and maximum probabilities of winning, the game is fair because if he picks one such die, you can pick the other; if one die is better than the other three, your opponent can always choose that die and win more than half of the contests. This reasoning is com-

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Nontransitive dice pletely wrong. The incredible truth is that regardless of which die he picks you can always pick a die that has a 2/3 probability of winning, or two-to-one odds in your favor!

The paradox (insofar as it violates common sense) arises from the mistaken assumption that the relation "more likely to win" must be transitive between pairs of dice. This is not the case with any of the three sets of dice. In each set the relation "more likely to win" is indicated by an arrow that points to the losing die. Die A beats B, B beats C, C beats D—and D beats Al In the first set the probability of winning with the indicated die of each pair is 2/3. This is easily verified by listing the 36 possible throws of each pair, then checking the 24 cases in which one die bears the highest number.

The other two sets of four dice, also designed by Efron, have the same nontransitive property but fewer numbers are repeated in order to make an analysis of the dice more difficult. In the second set the probability of winning with the indicated die is also 2/3. Because ties are possible with the third set it must be agreed that ties will be broken by rolling again. With this procedure the winning probability for each of the four pairings in the third set is 11/17, or .647.

It has been proved, Efron writes, that 2/3 is the greatest possible advantage that can be achieved with four dice. For three sets of numbers the maximum advantage is .618, but this cannot be obtained with dice because the sets must have more than six numbers. If more than four sets are used (numbers to be

Chapter s randomly selected within each set), the possible advantage approaches a limit of 3/4 as the number of sets increases.

A fundamental principle in calculating probabilities such as dice throws is one that goes back to the beginnings of classical probability theory in the 18th century. It was formerly called "the principle of insufficient reason" but is now known as "the principle of indifference," a crisper phrase coined by John Maynard Keynes in A Treatise on Probability. (Keynes is best known as an economist, but his book on probability has become a classic. It had a major influence on the inductive logic of Rudolf Carnap.) The principle is usually stated as follows: If you have no grounds whatever for believing that any one of n mutually exclusive events is more likely to occur than any other, a probability of 1 In is assigned to each.

For example, you examine a die carefully and find nothing that favors one side over another, such as concealed loads, noncubical shape, beveling of certain edges, stickiness of certain sides and so on. You assume that there are six equally probable ways the cube can fall; therefore you assign a probability of 1/6 to each. If you toss a penny, or play the Mexican game of betting on which of two sugar cubes a fly will alight on first, your ignorance of any possible bias prompts you to assign a probability of 1/2 to each of the two outcomes. In none of these samples do you feel obligated to make statistical, empirical tests. The probabilities are assigned a priori. They are based on symmetrical features in the structures and forces involved. The die is a regular solid, the probability of the penny's balancing on its edge is virtually zero, there is no reason for a fly to prefer one sugar cube to another and so on. Ultimately, of course, your analysis rests on empirical grounds, since only experience tells you, say, that a weighted die face would affect the odds, whereas a face colored red (with the others blue) would not.

Some form of the principle of indifference is indispensable in probability theory, but it must be carefully qualified and applied with extreme caution to avoid pitfalls. In many cases the traps spring from a difficulty in deciding on what are the equally probable cases. Suppose, for instance, you shuffle a packet of four cards—two red, two black—and deal them face down in a row. Two cards are picked at random, say by placing a penny on each. What is the probability that those two cards are the same color?

One person reasons: "There are three equally probable cases. Either both cards are black, both are red or they are dif ferent colors. In two cases the cards match, therefore the matching probability is 2/3."

"No," another person counters, "there are four equally probable cases. Either both cards are black, both are red, card x is black and y is red or % is red and y is black. More simply, the cards either match or they do not. In each way of putting it the matching probability clearly is 1/2."

The fact is that both people are wrong. (The correct probability will be given in the Answer Section. Can the reader calculate it?) Here the errors arise from a failure to identify correctly the equally probable cases. There are, however, more confusing paradoxes—actually fallacies—in which the principle of indifference seems intuitively to be applicable, whereas it actually leads straight to a logical contradiction. Cases such as these result when there are no positive reasons for believing n events to be equally probable and the assumption of equiprob-ability is therefore based entirely, or almost entirely, on ignorance.

For example, someone tells you: "There is a cube in the next room whose size has been selected by a randomizing device. The cube's edge is not less than one foot or more than three feet." How would you estimate the probability that the cube's edge is between one and two feet as compared with the probability that it is between two and three feet? In your total ignorance of additional information, is it not reasonable to invoke the principle of indifference and regard each probability as 1/2?

It is not. If the cube's edge ranges between one and two feet, its volume ranges between Is, or one, cubic foot and 23, or eight, cubic feet. But in the range of edges from two to three feet, the volume ranges between 23 (eight) and 33 (27) cubic feet—a range almost three times the other range. If the principle of indifference applies to the two ranges of edges, it is violated by the equivalent ranges of volume. You were not told how the cube's "size" was randomized, and since "size" is ambiguous (it could mean either the cube's edge or its volume) you have no clues to guide your guessing. If the cube's edge was picked at random, the principle of indifference does indeed apply. It is also applicable if you are told that the cube's volume was picked at random, but of course you then have to assign a probability of 1/2 to each of the two ranges from one to 14 and from 14 to 27 cubic feet, and to the corresponding ranges for the cube's edge. If the principle applies to the edge, it cannot apply to the volume without contradiction, and vice versa. Since you do not know how the size was selected, any application of the principle is meaningless.

Carnap, in attacking an uncritical use of the principle in Harold Jeffreys' Theory of Probability, gives the following example of its misuse. You know that every ball in an urn is blue, red or yellow, but you know nothing about how many balls of each color are in the urn. What is the probability that the first ball taken from the urn will be blue? Applying the principle of indifference, you say it is 1/2. The probability that it is not blue must also be 1/2. If it is not blue, it must be red or yellow, and because you know nothing about the number of red or yellow balls, those colors are equally probable. Therefore you assign to red a probability of 1/4. On the other hand, if you begin by asking for the probability that the first ball will be red, you must give red a probability of 1/2 and blue a probability of 1/4, which contradicts your previous estimates.

It is easy to prove along similar lines that there is life on Mars. What is the probability that there is simple plant life on Mars? Since arguments on both sides are about equally cogent, we answer 1/2. What is the probability that there is simple animal life on Mars? Again, 1/2. Now we are forced to assert that the probability of there being "either plant or animal life" on Mars is 1/2+ 1/2=1, or certainty, which is absurd. The philosopher Charles Sanders Peirce gave a similar argument that seems to show that the hair of inhabitants on Saturn had to be either of two different colors. Many variants of this fallacy can be found in Chapter 4 of Keynes's book. It is easy to invent others.

In the history of metaphysics the most notorious misuse of the principle surely was by Blaise Pascal, who did pioneer work on probability theory, in a famous argument that became known as "Pascal's wager." A few passages from the original and somewhat lengthy argument (in Pascal's Pensées, Thought 233) are worth quoting:

"God is, or he is not." To which side shall we incline? Reason can determine nothing about it. There is an infinite gulf fixed between us. A game is playing at the extremity of this infinite distance in which heads or tails may turn up. What will you wager? There is no reason for backing either one or the other, you cannot reasonably argue in favor of either. . . .

Yes, but you must wager. . . . Which will you choose? . . . Let us weigh the gain and the loss in choosing "heads" that God is. . . . If you gain, you gain all. If you lose, you lose nothing. Wager, then, unhesitatingly that he is.

Lord Byron, in a letter, rephrased Pascal's argument effectively: "Indisputably, the firm believers in the Gospel have a great advantage over all others, for this simple reason—that, if true, they will have their reward hereafter; and if there be no hereafter, they can be but with the infidel in his eternal sleep, having had the assistance of an exalted hope through life, without subsequent disappointment, since (at the worst for them) out of nothing nothing can arise, not even sorrow." Similar passages can be found in many contemporary books of religious apologetics.

Pascal was not the first to insist in this fashion that faith in Christian orthodoxy was the best bet. The argument was clearly stated by the fourth-century African priest Arnobius the Elder, and non-Christian forms of it go back to Plato. This is not the place, however, to go into the curious history of defenses and criticisms of the wager. I content myself with mentioning Denis Diderot's observation that the wager applies with equal force to other major faiths such as Islam. The mathematically interesting aspect of all of this is that Pascal likens the outcome of his bet to the toss of a coin. In other words, he explicitly invokes the principle of indifference to a situation in which its application is mathematically senseless.

The most subtle modern reformulation of Pascal's wager is by William James, in his famous essay The Will to Believe, in which he argues that philosophical theism is a better gamble than atheism. In a still more watered-down form it is even used occasionally by humanists to defend optimism against pessimism at a time when the extinction of the human race seems as likely in the near future as its survival.

"While there is a chance of the world getting through its troubles," says the narrator of H. G. Wells's little read novel Apropos of Dolores, "I hold that a reasonable man has to behave as though he was sure of it. If at the end your cheerfulness is not justified, at any rate you will have been cheerful."

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