In a minute, earn cash in your spare time! Bar bets you can win. But first . . .

THE MONTY HALL PROBLEM

As clear as it was to me that your odds of winning the mega-yacht could not be improved by switching from your first choice, it’s now equally clear to me – now that I understand it – that I was wrong. Beyond dispute.

David Rothman and Keith Skilling were among the many of you who really nailed it for me: Imagine not 3 doors but a million. You choose door number 220601(say). The game show host opens 999,998 doors leaving just yours (number 220601) and one other. Now do you want to stick with the door you choose, figuring that you hit a million-to-one shot with your first pick? Or would you prefer to switch to the only door besides yours I left unopened?

Here the answer is intuitively obvious. Play the game with just 3 doors instead of a million and it’s the same principle, just a bit more subtle.

What’s fascinating, I think (and a little scary), is that so many math professors could have been equally certain – and wrong – as reported years ago in a New York Times conversation with Monty hall himself.

Geoff Townsend:Here is an elegant solution to the problem, making it seem almost intuitive. The author, Jeffrey Rosenthal, a statistics professor, has also written an interesting book on everyday probabilities, Struck by Lightning: The Curious World of Probabilities.’

Assume that you always start by picking Door #1, and the host then always shows you some other door which does not contain the car, and you then always switch to the remaining door.

If the car is behind Door #1, then after you pick Door #1, the host will open another door (either #2 or #3), and you will then switch to the remaining door (either #3 or #2), thus LOSING.

If the car is behind Door #2, then after you pick Door #1, the host will be forced to open Door #3, and you will then switch to Door #2, thus WINNING.

If the car is behind Door #3, then after you pick Door #1, the host will be forced to open Door #2, and you will then switch to Door #3, thus WINNING.

Hence, in 2 of the 3 (equally-likely) possibilities, you will win. Ergo, the probability of winning by switching is 2/3.

Gray Chang: ‘An important part of the Monty Hall problem is often omitted from the problem statement. Monty knows which door holds the prize and he always reveals a clunker door. Then you should indeed switch your choice, as you said. However, it’s a different situation if Monty doesn’t know which door holds the prize and he reveals one of the two remaining doors at random. In that case, one-third of the time he will reveal the prize door, and you lose the game. Two-thirds of the time he will reveal a clunker, in which case there is no advantage or disadvantage of changing your original choice; the odds are 50-50 at that point.

Michael Young: ‘One absolutely critical part of the problem, that nobody ever mentions, is whether Monty’s rules required him to show you one of the losing doors. If he has the option not to show you a door at all, then you simply cannot make any inferences, and your ‘intuition’ that switching should make no difference is entirely correct – you’re playing a mind game with Monty. (Consider that without this rule, he could choose to show you a door only when he knows you have chosen the winning door already. Or not, to fake you out.) I was too young when Monty was doing his thing to have paid attention to whether Monty ever refrained from offering another door.’

Sergei Slobodov: ‘By forcing the host to choose the one yacht-free door of the two doors remaining, you are taking advantage of his knowing more than you do (after all, he will never open the door with the yacht behind it). In trading terms, he is an insider and you are taking advantage of his inside information!

George Hamlett: ‘You write: ‘One’s grasp for related knowledge immediately goes to the coin toss truism: that even if a coin has come up ‘tails’ ten times in a row, a cool-headed man or woman knows it is no more likely to come up ‘heads’ on the eleventh. The odds of an honest coin-toss are 50-50 every time.’ That’s true, of course, but the key word is honest. What are the odds that any coin that comes up the same ten times in a row is an honest coin? Not good. So in a real-life situation, the right bet is tails, because the odds are it’s not an honest coin. Mr. ‘Black Swan,’ Nassim Nicholas Taleb, writes about this very problem of not confusing academic situations with the real world.’

And now . . .

BAR BETS YOU CAN WIN

I said this Monty Hall thing had no value except perhaps in winning bar bets. (Better still, bar bets with cocksure math majors.) But that produced other money-making opportunities.

Peter Baum: ‘Your mention of bar bets brings back some fond memories. Back in my youth I was an . . . ‘independent entrepreneur specializing in extremely short-time transactions based on psychological dislocation and information asymetry’ (hustler). Here‘s a youtube link to one of the classics, the Five Questions game. He’s using it to try to pick up a girl, but it can be used at least as effectively to win money. Enjoy.’

Dan Nachbar: ‘My favorite bar bet – Rome is further north than New York City. (It is.)’

☞ Double or nothing? Which is further north, Venice or Bangor, Maine?’ I win again! (Or use Fargo, North Dakota.)

What is the westernmost state in the Union? (Wrong!) The easternmost? (Wrong!) Oh, I like this.

(In both cases the answer is the same: Alaska. Which is also the northernmost. Hawaii is southernmost.)

What we clearly need to do to balance our trade deficit and strengthen the dollar is (a) attract more wealthy tourists; and (b) get them into the bars with us.*

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*The only possible glitch: foreigners actually learn geography. Uh, oh.**
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**Between yesterday and today, it seems to be footnote week.***
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***Sorry! I am really only 12 years old – and it shows.****
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****Nanotechnology! A whole computer could fit inside the period at the end of this sentence.

 

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