When intuition and math probably look wrong

A twist on the Two Children Problem shows how information can steer what looks probable

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

WORKING IT OUT This grid gives a visual way of calculating the answer to the question, “Among families that have two children, one of whom is a boy born on a Tuesday, how many have two boys?” It shows all the possible variations of birth day and gender for children in a two-child family. The families with the saturated colors are those which might qualify, while those that are shaded out are eliminated. The row gives the birth day of the older child and the column gives the birth day of the younger, while the color represents the sex of each child. If the older child is a boy born on Tuesday, then the family must lie on the Tuesday row and the box representing the older child must be blue. If the younger child is a boy born on Tuesday, the family must lie on the Tuesday column and the box representing the younger child must be blue. The combination of all those boxes show all the possible families that qualify. The total number of saturated boxes is 27. The number of saturated boxes in which both children are boys is 13. So the probability is 13/27. Bill Casselman
VISUAL CALCULATION This grid gives a visual way of calculating the answer to the question, “Among children with a single sibling, how often is that sibling of the same sex? And oh, by the way, mention the sex and birth day of the selected child along the way.” This is the interpretation that Peres argues is the most natural one in real life. Because the sex and birth day don’t influence the selection of the family, any two-child family would qualify, and in half of them, both are boys or both are girls. Bill Casselman

Gary Foshee, a puzzle designer from Issaquah, Wash., posed this puzzle during his talk this past March at Gathering 4 Gardner, a convention of mathematicians, magicians and puzzle enthusiasts held biannually in Atlanta. The convention is inspired by Martin Gardner, the recreational mathematician, expositor and philosopher who died May 22 at age 95. Foshee’s riddle is a beautiful example of the kind of simple, surprising and sometimes controversial bits of mathematics that Gardner prized and shared with others.

“The first thing you think is ‘What has Tuesday got to do with it?’” said Foshee after posing his problem during his talk. “Well, it has everything to do with it.”

Even in that mathematician-filled audience, people laughed and shook their heads in astonishment.

When mathematician Keith Devlin of Stanford University later heard about the puzzle, he too initially thought the information about Tuesday should be irrelevant. But hearing that its provenance was the Gathering 4 Gardner conference, he studied it more carefully. He started first by recalling a simpler version of the question called the Two Children Problem, which Gardner himself posed in a Scientific American column in 1959. It leaves out the information about Tuesday entirely: Suppose that Mr. Smith has two children, at least one of whom is a son. What is the probability both children are boys?

Intuition would suggest that the answer should be 1/2, since the sex of one child is independent of the sex of the other. And indeed, had he been told which child was a boy (say, the younger one), this reasoning would be sufficient. But since the boy could be either the younger or the older child, the analysis is more subtle. Devlin started by listing the children’s sexes in the order of their birth:

Boy, girl
Boy, boy
Girl, boy

Since one child is a boy, we know that girl, girl isn’t a possibility. Of the three approximately equally likely possibilities, one has two boys and two have a girl and a boy — so the probability of two boys is 1/3, not 1/2, Devlin concluded.

He used this same method on the Tuesday birthday puzzle, enumerating the equally likely possibilities for the sex and birth day of each child and then counting them up.

If the older child is a boy born on Tuesday, there are 14 equally likely possibilities for the sex and birth day of his younger sibling: a girl born on any of the seven days of the week or a boy born on any of the seven days of the week. (This analysis ignores minor differences like the fact that slightly more babies are born on weekdays than on weekend days.)

Now suppose that the older child isn’t a boy born on Tuesday. The younger child then must be, of course. Now we count up the possibilities for the sex and birth day of the older child. If she’s a girl, she might have been born on any day of the week, generating seven more possibilities. If he’s a boy, he could have been born any day except Tuesday. (Otherwise this case would already have been counted in the first scenario: the older child a boy born on Tuesday). This second scenario generates just six, rather than seven, more possibilities.

Since each of these cases is (approximately) equally likely, we can compute the probability by dividing the number of cases in which there are two boys by the total number of cases. The total number of cases is 27: 14 if the older child is a boy born on Tuesday and 13 if the older child isn’t. In 13 of those cases both children are boys (7 if the older child is a boy born on Tuesday and 6 if he isn’t), yielding a probability of 13/27.

Devlin was astonished by this answer. As a mathematician, he had long been familiar with the Two Children Problem and its answer of 1/3. “Knowing the birth day is a Tuesday may (and does) make a difference, but it surely cannot make much of a difference, right?” he wrote in his blog, Devlin’s Angle. “Wrong.” After all, 13/27 is far closer to 1/2 than 1/3.

So why does intuition seem to lead us so astray? Both the intuitive and the mathematically informed guesses are wrong. Are human brains just badly wired for computing probabilities?

Not so fast, says probabilist Yuval Peres of Microsoft Research. That naïve answer of 1/2? In real life, he says, that will usually be the most reasonable one.

Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2. The child’s sex and birthday are just information offered after the selection is made, which doesn’t affect the probability in the slightest.

Gardner himself tripped up on his simpler Two Children Problem. Initially, he gave the answer as 1/3, but he later realized that the problem is ambiguous in the same way that Peres argues that the Tuesday Birthday Problem is. Suppose that you already knew that Mr. Smith had two children, and then you meet him on the street with a boy he introduces as his son. In that case, the probability the other child is a son would be 1/2, just as intuition suggests. On the other hand, suppose that you are looking for a male beagle puppy. You want a puppy that has been raised with a sibling for good socialization but you are afraid it will be hard to select just a single puppy from a large litter. So you find a breeder who has exactly two pups and call to confirm that at least one is male. Then the probability that the other is male is 1/3.

In the scenario of Mr. Smith, you’re randomly selecting a child from his two children and then noticing his sex. In the puppy scenario, you’re randomly selecting a two-puppy family with at least one male.

The remarkable thing that Foshee’s variation points out is that any piece of information that affects the selection will also affect the probability. If, for example, you selected a family at random among those with two kids, one of whom is a boy who plays the ukulele and wants to become a dancer, the ukulele-playing and dancing ambitions would affect the probabilities about the sex of his sibling.

Peres says that we shouldn’t despair about our probabilistic intuition, as long as we apply it to familiar situations. The difficulty of these problems is rooted in their artificiality: In real life, we almost always know why the information was selected, whereas these problems have been devised to eliminate that knowledge. “The intuition develops,” he points out, “to handle situations that actually occur.”

Still, Gardner’s initial overly narrow interpretation warns of the dangers of over-hasty analysis of probability questions — and shows the wonder that can come from them.

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