Editor’s Note: Published in 1957, this article comes from Martin Gardner’s legendary Scientific American column Mathematical Games. Read more in our special digital issue, Fun and Games.
A paradox is a truth which cuts so strongly against the grain of common sense that it is hard to believe even when you are confronted with the proof. This quality of incredibility is particularly true of paradoxes in probability—a field of mathematics especially rich in paradoxes.
Consider the paradox of birthdays. What would you estimate to be the probability that, in any group of 24 persons, two or more were born on the same day of the same month? Offhand you would say it will be very low. In fact, the probability is 27/50, or better than one half! In other words, if you were to bet even money on there being at least one coincidence of birthdays in a random collection of 24 persons, you would have a better than even chance of winning—over the long run.
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These odds are so unexpected that you can make an entertaining, as well as profitable, game of the thing at parties or other gatherings of 24 or more people. Let each person write his birthday on a slip of paper. More often than not, at least two of the birthdays will be the same—sometimes to the surprise of the parties concerned, though they may have known each other for years.
You don’t need a party of 24 to play the game: you can merely take 24 names at random out of Who’s Who or some other biographical dictionary. I looked up the birthdays of the 33 Presidents of the U.S. and am happy to report that they obeyed the law of averages. Two Presidents had the same birthday: James Polk and Warren Harding were born on November 2.
The calculation of these odds from probability principles is perfectly simple but rather tedious. One method of calculating them is given by George Gamow in his book One Two Three ... Infinity. The probability that the birthdays of two persons will not coincide is 364/365, since there are 364 chances in 365 of their birthdays being different. The probability that a third person will have a birthday different from the first two is 363/365; for a fourth person the probability of a still different birthday is 362/365, and so on to 342/365 for the 24th person. To compute the probability that all 24 persons have different birthdays, you multiply all these probabilities together, and the result is a fraction which reduces to 23/50. This means that you would win 27 out of every 50 bets on a coincidence of birthdays in groups of 24 persons.
Even more startling is the paradox of the second ace. Suppose that a bridge player were to look at his freshly dealt hand and announce: “I have an ace.” What is the probability that he also has a second ace? This can be calculated precisely, and it proves to be a little under 1/3. But suppose he announced that he had a particular ace, say the ace of spades, selected by agreement in advance of the deal. The probability that the player holding the ace of spades also had another ace would be 11,686/20,825, or slightly better than 1/2! Why should naming the ace affect the odds?
To simplify the work of computation, we can illustrate the situation with a more elementary game using only four cards—the ace of spades, the ace of hearts, the deuce of clubs and the jack of diamonds. If two cards are dealt to each of two players, there are only six possible combinations that a player can hold: (1) ace of spades and ace of hearts, (2) ace of spades and jack of diamonds, (3) ace of spades and deuce of clubs, (4) ace of hearts and jack of diamonds, (5) ace of hearts and deuce of clubs, (6) jack of diamonds and deuce of clubs. Now five of these six two-card hands permit the player to say, “I have an ace,” and in one of the five instances he has a second ace. Consequently in this game the probability of the second ace is 1/5. But observe that if the player is able to declare that he holds the ace of spades, the probability that the second ace is in his hand goes up to 1/3, because there are only three combinations containing the ace of spades and one includes the second ace.
The most famous of all probability paradoxes is the St. Petersburg paradox, first set forth in a paper by the famous mathematician Daniel Bernoulli before the St. Petersburg Academy. Suppose I toss a penny and agree to pay you a dollar if it falls heads. If it comes tails, I toss again, this time paying you two dollars if the coin is heads. If it is tails again, I toss a third time and pay four dollars if it falls heads. In short, I offer to double the penalty with each toss and I continue until I am obliged to pay off. What should you pay for the privilege of playing this one-sided game with me?
The unbelievable answer is that you could pay me any amount, say a million dollars, for each game and still come out ahead. In any single game there is a probability of 1/2 that you will win a dollar, 1/4 that you will win two dollars, 1/8 that you will win four dollars, and so on. Therefore the total you may expect to win is (1 x 1/2) + (2 x 1/4) + (4 x 1/8).... The sum of this endless series is infinite. As a result, no matter what finite sum you paid me in advance per game, you would win in the end if we played enough games. You would be paid something in every game and you would also have a chance, albeit small, of winning an astronomical sum each time the game was played. This paradox is involved in every “doubling” system of gambling. Its full analysis leads into all sorts of intricate byways.
Carl C. Hempel, a leading figure in the “logical positivist” school and now a professor of philosophy at Princeton University, discovered another astonishing probability paradox. Ever since he first explained it in 1937 in the Swedish periodical Theoria, “Hempel’s paradox” has been a subject of much pleasant and learned argument among philosophers of science, for it reaches to the very heart of scientific method.
Let us assume, Hempel began, that a scientist wishes to investigate the hypothesis “All crows are black.” His research consists of examining as many crows as possible. The more black crows he finds, the more probable the hypothesis becomes. Each black crow can therefore be regarded as a “confirming instance” of the hypothesis. Hempel asserted that the existence of a brown stone also is a “confirming instance” of the hypothesis! He proved his paradox with ironclad logic.
The statement “All crows are black” can be transformed to the logically equivalent statement, “All not-black objects are not-crows.” The second statement is identical in meaning with the original. Consequently the discovery of any object that “confirms” the second statement must also confirm the first.
Suppose, then, that the scientist, searching about for not-black objects, comes upon a brown stone. This object is a confirming instance of “All not-black objects are not-crows.” It therefore must add to the probable truth of the equivalent hypothesis “All crows are black.” The same applies to a white elephant, or a red herring, or the scientist’s green necktie. As one philosopher recently remarked, on rainy days an ornithologist investigating the black-crow hypothesis could carry on his research without getting his feet wet. He need only explore his room and note instances of not-black objects that are not-crows!
We find it hard to accept the validity of this paradox, says Hempel, because of a “misguided intuition.” But it begins to make sense when we consider a simpler problem. Let us say that we wish to test the hypothesis that all red-haired typists working for a certain large company are married. We could investigate this directly by seeking out every red-haired typist and asking her if she has a husband. But there is another test, which might actually be more efficient. We could get a list of all the unmarried typists in the company from the personnel department and then investigate whether any of the girls on this list has red hair. If it turns out that no unmarried typist has red hair, we have completely confirmed our hypothesis that all of the red-headed typists are married. And each not-married typist with not-red hair serves effectively as a confirming instance of the hypothesis. If there are fewer unmarried than married typists, we could save time by this approach.
The only real difference between the problem of the red-headed typists and the one of the black crows is in the relative sizes of the classes. There are so many not-black objects in the world that checking them would be an extremely inefficient method of testing the hypothesis that all crows are black. Nonetheless most logicians agree that Hempel’s logic is unassailable. And although we may be tempted to dismiss Hempel’s paradox with a smile and a shrug, we must remember that many logical paradoxes which were long regarded as mere mental exercises proved to be highly useful in the development of symbolic logic. Analyses of Hempel’s paradox have already provided valuable insights into the obscure nature of inductive logic, the tool by which all scientific knowledge is obtained.