So You Think You've Got Problems by Alex Bellos

Author:Alex Bellos [Bellos, Alex]
Language: eng
Format: epub
ISBN: 9781783351923
Publisher: Guardian Faber Publishing
Published: 2019-01-15T05:00:00+00:00

[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?

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 a twist on Gardner’s two-boy problem in which Mr Smith has two children, one of whom was 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 than he is likely to have been born on any other day?

The Starbirds did not realise quite how their Tuesday-boy problem would cause 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 counter-intuitive 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.’ He wonders if people get emotionally involved in this type of puzzle because it challenges their basic mathematical confidence – ‘there may be a subconscious aspect that is upsetting, along the lines of: if my intuition is so wrong for such an easy-to-state problem, am I making errors all the time in other areas?’

The solution to (and further discussion of) the Tuesday-boy problem is provided in the answer to the next question.

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