False premises, patchy logic, right conclusion
In all my 49 years I don't think I've ever read a piece with so many hugely embarrassing basic errors which still staggers amusingly to a correct conclusion as the piece by Rod Liddle in this week's Spectator, under the title "Monty Hall will change the way you think."
I thought at first that Mr Liddle was making an ironic joke when he argued that what day of the week one sibling in a two-child family is born on affects the gender of the other sibling.
Of course it doesn't: Liddle has messed up a gratuitously complicated conditional probability calculation. The same sort of mistake which Professor Sir Roy Meadow made when he didn't realise that being one of Britain's leading experts in paediatric medicine did not automatically confer any understanding of statistics, leading him to persuade juries to send innocent women to jail for murders which never happened.
But the real joke is that although some of the details of Rod Liddle's article are complete rubbish, this merely provides more evidence that his basic argument is right: even very clever people can make disastrous errors through not understanding mathematical and statistical problems, especially when intuition leads people in false directions.
The "Monty Hall" paradox (which Liddle describes correctly) is an example of a case where the correct answer to a question appears at first to be absurd.
Monty Hall was the original host of a famous american game show, and the paradox is named after him because the problem was originally set out in a letter to the "American Statistician" in 1975 which used games shows as the example.
The idea is that there are three doors: behind one of the three is a prize, behind the other two is something worthless. The contestant is asked to make an initial selection of a door, but the door is left open. The host then opens one of the other two doors, revealing a booby prize, and asks the contestant if he or she wants to open the door originally selected, or switch to open the third door instead.
Because one door has a valuable prize behind it and the other a worthless one, there is an instinctive tendancy to assume that the chance of finding the valuable prize is 50:50 whichever door was opened. When the puzzle was first unveiled, thousands of people who ought to have known better insisted that this must be the case.
And if the host's decision was based on random chance, and Monty Hall just happened to open a door which had a booby prize behind it, this is quite correct.
But if the host KNEW which door had the real prize, and deliberately selected a door with a worthless prize based on that knowledge, switching to the third door doubles your chance of a correct answer, from one third to two-thirds.
This is because the contestant made the original selection from among three options, and therefore there was (and is) a one-third chance of the first door chosen being a correct one. When the host opens the second door and knowingly eliminates a wrong answer, that leaves the third door which must have the remaining probability of hiding the valuable prize - a two thirds chance.
Is your head spinning? Don't worry, you're in good company.
By chance I myself have recently been reading one of the two books which Rod Liddle recommends as a way to avoid confusing yourself by spotting non-existent patterns. It's called "The Tiger That Isn't: Seeing Through a World of Numbers" by Michael Blastland and Andrew Dilnot, and I can endorse his recommendation.
I thought at first that Mr Liddle was making an ironic joke when he argued that what day of the week one sibling in a two-child family is born on affects the gender of the other sibling.
Of course it doesn't: Liddle has messed up a gratuitously complicated conditional probability calculation. The same sort of mistake which Professor Sir Roy Meadow made when he didn't realise that being one of Britain's leading experts in paediatric medicine did not automatically confer any understanding of statistics, leading him to persuade juries to send innocent women to jail for murders which never happened.
But the real joke is that although some of the details of Rod Liddle's article are complete rubbish, this merely provides more evidence that his basic argument is right: even very clever people can make disastrous errors through not understanding mathematical and statistical problems, especially when intuition leads people in false directions.
The "Monty Hall" paradox (which Liddle describes correctly) is an example of a case where the correct answer to a question appears at first to be absurd.
Monty Hall was the original host of a famous american game show, and the paradox is named after him because the problem was originally set out in a letter to the "American Statistician" in 1975 which used games shows as the example.
The idea is that there are three doors: behind one of the three is a prize, behind the other two is something worthless. The contestant is asked to make an initial selection of a door, but the door is left open. The host then opens one of the other two doors, revealing a booby prize, and asks the contestant if he or she wants to open the door originally selected, or switch to open the third door instead.
Because one door has a valuable prize behind it and the other a worthless one, there is an instinctive tendancy to assume that the chance of finding the valuable prize is 50:50 whichever door was opened. When the puzzle was first unveiled, thousands of people who ought to have known better insisted that this must be the case.
And if the host's decision was based on random chance, and Monty Hall just happened to open a door which had a booby prize behind it, this is quite correct.
But if the host KNEW which door had the real prize, and deliberately selected a door with a worthless prize based on that knowledge, switching to the third door doubles your chance of a correct answer, from one third to two-thirds.
This is because the contestant made the original selection from among three options, and therefore there was (and is) a one-third chance of the first door chosen being a correct one. When the host opens the second door and knowingly eliminates a wrong answer, that leaves the third door which must have the remaining probability of hiding the valuable prize - a two thirds chance.
Is your head spinning? Don't worry, you're in good company.
By chance I myself have recently been reading one of the two books which Rod Liddle recommends as a way to avoid confusing yourself by spotting non-existent patterns. It's called "The Tiger That Isn't: Seeing Through a World of Numbers" by Michael Blastland and Andrew Dilnot, and I can endorse his recommendation.
Comments
If it is true for 3 doors then it should also be true for 100 doors. So imagine this 100 door example and you are down to the last 2 doors, does switching choices really give you a 99% chance of picking the correct door?
Yes, Anonymous, I checked it. You can find a more detailed explanation at Wikipedia.
And yes, if you had a hundred doors, one with a real prize behind, and a host who knew which door that was opened 98 of the remaining 99 after the contestant had made a initial selection, to reveal 98 booby prizes: then there would be one chance in 100 that the prize would be behind the door originally chosen, and 99 that it would be behind the one remaining door.