Tuesday, 22 June 2010

Tuesday Boy

You're giving me credit for being more of a mathematician than I really am. I'm strictly a journalist. I just write about what other people are doing in the field." (Martin Gardner)

Martin Gardner, who has died recently aged 95, was considerably too modest. With his mathematical puzzles and games, he entertained generations, and recreational mathematics was a bridge to drawing people into all kinds of deep mathematical theory, with fun rather than the pain so many people remember of their school experience of maths. And he was a man full of fun - the Telegraph reports how "he once wrote a devastating review of one of his own books and got it published under a pseudonym in the New York Review of Books".

The obituaries also mention that he was a sceptic, albeit a "philosophical theist", and published "The Annotated Alice", in which he delves behind the text to show Lewis Carroll's inspiration in puzzles, and spoofs of well know (but now almost forgotten) nursery songs and rhymes of the Victorian age. We learn how the Cheshire Cat got its grin, why the Mock Turtle wept, and who inspired the Walrus and the Carpenter, and the meanings that the curious words in Jabberwocky may have.

Yet it is not dry academic notes, but an inspiring annotation which enhances the original text. It was - and is - so popular that he also annotated Carroll's lesser known masterpiece of existential angst, "The Hunting of the Snark".

For someone with an antipathy to more organised religion, it was strange that he also found a kindred spirit in Chesterton, producing "The Annotated Innocence of Father Brown", and an introduction to "The Man Who Was Thursday".

You meet someone who tells you, "I have two children. One of them is a boy. What is the probability I have two boys?"

On hearing this for the first time many people think the answer must be 1/2. There are two children. One is a boy. So the other must be a boy or a girl with equal probability. But the mathematical trap lurking in the question is we are told there are two children, and one of them (but we don't know which) is a boy. It is like tossing two coins together, and then revealing one is heads. The possibilities are:

Boy Girl, Girl Boy, Boy Boy or Girl Girl

But we know one is a boy, which leaves just

Boy Girl, Girl Boy, Boy Boy

So the chance of Boy Boy being right, and a Boy being the unseen child, must be 1/3

Now there is a recreational mathematics symposium called "Gathering 4 Gardner", named in honour of Martin Gardner. In 2009, puzzle-maker Gary Foshee got on stage and proposed a problem that is like that, but even more tricky:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? Your first impression is: what does Tuesday have to do with it?" says Gary, "And you might think that it doesn't. But in fact Tuesday has everything to do with it. "

For the solution, and more beside, read the details on BBC Radio 4's "More or Less" website at:

1 comment:

Richard Jowitt said...

This reasoning is entirely flawed but has achieved widespread acceptance on the web. The mistake stems from confusing the difference between combinations (without regard to order/sequence) and permutations (where the order defines a different state). The possible combinations of 2 children, without regard to order are: 2 boys, 2 girls or one of each. We know that there is at least one boy so the probability of the combination being 2 boys is simply 1/2. If the analysis is made using permutations, where girl+boy is different from boy+girl, then the boy+boy permutations must also be counted as two different states i.e. boy(a)+boy(b) and boy(b)+boy(a). This is the subtle distinction which is commonly overlooked and which keeps the probability of 2 boys at 1/2. When these 'additional' possibilities are included in the full weekday analysis the result is not 13/27 but 14/28 i.e. 1/2.