My, what a long explanation... Here's an alternative.

Imagine that instead of three doors there are ten thousand of them. One's got the car, the rest have got goats. You choose one. Monty opens 9998 doors (this is a really long and boring game show!), all of which he knows have goats behind them, and asks if you want to switch.

When you picked a door you had one chance in ten thousand of getting the right one. That means that there's a 99.99% chance that the car is behind one of the 9999 doors you didn't pick. And now after Monty has told you about 9998 wrong doors, there's still a 99.99% chance that the car is behind a door you didn't chose -- only now there's only one such door left to choose from.

So, do you switch or not?

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Yes, someone raises that point in the crookedtimber discussion, and I think it makes it easier to udnerstand.

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I'd prefer a goat anyway.....

Here's another maths puzzle I have never understood:

Three men go out to dinner. The bill, when they get it, comes to £30 exactly - what could be more convenient, everyone pays £10 each and gives it to the waiter. He takes it to the cashier, who checks and finds a mistake - the bill should be £25. She gives the waiter 5 pound coins to take back to the table.

The waiter, being dishonest, pockets two and takes three back. He explains the overcharge and each man takes back one pound coin.

Each has now paid £9. Three nines are 27. The waiter has £2: 27 and 2 makes 29.

Where's the other pound?

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The puzzle is deliberately misleading. You need to count down, not up; 'thirty minus three is twenty-seven, minus two is twenty-five.' Or to put it another way:

At the start, the restaurant should have 25, but actually has 30.

At the end, the restaurant has 25, the waiter has 2, and each customer has 1.

At the start, the customers have paid 10, but should have paid 8.33.

At the end, the customers have paid 9, but should have paid 8.33.

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There's a good--and much briefer and simpler--explanation of the solution (which is to switch) in the book The Curious Incident of the Dog in the Night by Mark Haddon. The relevant excerpt is here.

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*smacks forehead* I knew I'd read about this recently! Thanks!

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Thanks! My daughter has that book - I ought to read it

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It's well worth it. And hey, B7 gets a brief mention!

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It must have been 15 years or so ago when I first came across this problem. I still remember the (whole) afternoon I spent refusing to believe the answer, until eventually I got it.

Someone (the Saint ?) uses this problem to swindle a bad guy out of lots of money. He has three sticks - one painted red at both ends, one green at both ends, and one which has one green and one red end. You pick a stick at random and show one end, and your opponent has to guess what the concealed colour is. You're supposed to think it's 50:50 but really it's 2/3 that the concealed colour is the same as the colour showing. The hero knows the bad guy will work this out, so he uses sleight of hand to reverse the odds and nab his dosh.

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Do you remember the discussion we had on livejournal a few months ago about false positives and probability of tests givign right answers? I've discovered it's called 'Bayesian analysis'. I'm sure you already knew. It turns out there's a whole industry out there discussing it.

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Not sure I remember the discussion, so sorry if the following is repeating that. I don't know too much about Bayesian stuff, but I understand that one of the most interesting/worrying things they say goes like this:

Person gets accused of crime. Forensic evidence is found at the scene which matches suspect. Prosecution says something like "the probability of a match is one in a million". Instead of treating this as compelling evidence of guilt, what we ought to be thinking about is the other 60 people in the country who would also have matched.

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Yes, or if you had a background check which detected 100% of terrorists, and had a 0.5% chance of incorrectly identifying an innocent person as a terrorist, that would sound pretty good, but it would result in thousands and thousands of false arrests to every one safe conviction.

Possibly it was a different maths dude I was talking to about this. However, now I have 'got it' the application and importance of the idea keeps coming to my attention.

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I came up with a different answer:

http://www.tallent.us/CommentView.aspx?guid=e24dfa09-e103-4e2f-bd85-eac6832b11e5

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Hi - I read your blog, and I think I concur with Ian G's comments. But it's an interesting debate in any case. Try the simulation, which I linked to, which demonstrates the frquency of correct answers.

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