Perilous Problems for Puzzle Lovers by Alex Bellos

Author:Alex Bellos
Language: eng
Format: epub
Publisher: The Experiment
Published: 2020-10-07T19:28:09+00:00

Albert has exactly two children, the oldest of which is a son.

Is a reader of the newspaper correct to deduce that the chance of Albert having two sons is 1 in 2?

[3] Beth has two children.

You: Can you think of one of your children?

Beth: OK, I have one in mind.

You: Is that child a girl?

Beth: Yes.

What do you estimate are the chances that Beth has two girls?

[4] All you know about Caleb is that he has two children, at least one of whom is a girl. We can imagine asking him whether his older child is a girl, or whether his younger child is a girl, and we know his answer to at least one of these questions will be “yes.” Does this insight alter the chances of his having two girls from 1 in 3 to1 in 2?

[5] A year ago, I learned that Caleb has exactly two children. I asked him one of the following two questions: “Is your older child a girl?” or “Is your younger child a girl?” I know that his answer was “yes,” but I can’t remember which question I asked! It’s a 50/50 chance I asked either question. What are the chances he has two girls?

All these problems were set by the brothers Tom and Michael Starbird.

(Question to Mrs. Starbird: You have two sons. At least one is a boy. What is the probability that both will become mathematicians?)

Tom is a veteran NASA scientist who has worked on many space missions and continues to help operate the Mars Curiosity rover, while Michael is a professor of mathematics at the University of Texas.

Michael also presents home-learning lecture courses. In the early 2000s, when he was preparing one on probability, the brothers came up with their twist on Gardner’s two-boy problem (in which Mr. Smith has two children, one of whom is a boy born on a Tuesday). Remarkably, once you mention the day of the week, the probability of Mr. Smith having two boys is no longer 1⁄2 or 1⁄3 but somewhere in between (assuming that Mr. Smith is randomly chosen from the population of all two-children families). The answer seems incomprehensible. How on Earth can knowing the day of the week make a difference to the probability that both children are boys, since the boy under discussion is equally likely to have been born on a Tuesday as he is likely to have been born on any other day?

The Starbirds did not realize quite how their Tuesday–boy problem would leave a trail of misery and irritation. “Many people resist the idea that apparent irrelevancies can have an impact on probabilities,” says Michael. “People are so upset by it. Of course, surprises and counterintuitive results are good, so I like people to be startled, but in this case the ‘aha’ moment sometimes never comes.”

Tom says that two of his colleagues at the Jet Propulsion Laboratory reacted to the puzzle with “close to genuine anger. There is something about the puzzle that is seriously disconcerting to some people.

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