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.)”
F 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.*
_________
*The
only possible glitch: foreigners actually learn geography. Uh, oh.**
___________
**Between yesterday and
today, it seems to be footnote week.***
____________
***Sorry! Iam really only 12
years old – and it shows.****
________________
****Nanotechnology! A whole computer could fit inside the period at
theend of this sentence.