Re: Thinking Probabilistically

From: Stevan Harnad (
Date: Mon Feb 12 1996 - 17:05:33 GMT

> From: "EMMA FLETCHER" <>
> Date: Mon, 12 Feb 1996 15:50:09 GMT
> Stevan, I'm still having problems with some of the
> statistics we discussed, especially the game show bit with the three
> curtains... why change? I don't understand why the odds change as
> they do. Why isn't it 50/50 between the remaining two? ( I know that
> you have explained this already but I just can't grasp it, sorry).

Here's how the odds change: Remind yourself that the revealing
of the empty curtain is giving you NEW INFORMATION that you don't
have when you just see the three curtains. Information reduces
uncertainty; it changes the odds. When there are three curtains, your
chance of being right is 1/3 -- and if you do not use any new
information you get later, then that's exactly what it stays: 1/3. One
third of the time, your first choice will be right, by chance. No more,
no less. (You must always think of doing this over and over again; it is
the long run that shows clearly which strategy is right.)

Now AFTER you have chosen, and REVEALED your choice, the person who
KNOWS where the prize is (and which curtain you have chosen) shows you
one empty curtain from out of the two you DIDN'T choose. It's always
one that you didn't choose, so that IS information. It's as if he now
said: choose from these TWO (the one you choose and the one I haven't
shown to be empty). Forget about the third one: it's empty.

Now you agree that if he did that BEFORE you had chosen, every time,
then your chances would be 1/2: You'd just be choosing one out of two.
But if he did it before you chose, he would sometimes reveal that the
one you WOULD HAVE chosen was empty. (To prove this, choose one mentally
before he reveals an empty one, and you'll find that 1/3 of the time the
empty one he reveals will be the one you chose. So in THAT case it's
OBVIOUS why it's better to change your choice.)

So let's see what we have so far: You see that if you stick with your
first choice, you win 1/3 of the time. You see that if you toss a coin
to decide which of the two remaining curtains to choose after he has
revealed an empty one, you win 1/2 the time. Can you do better than

Yes, but now reckon on the odds AGAINST you: Once you choose the first
time, 2/3 of the time the prize will NOT be behind the curtain you
chose. In other words, 2/3 of the time it will be behind one or the
other curtain you did NOT choose. Now if someone PROVES that it isn't
behind ONE of the two you didn't choose, that means that (2/3 of the
time, still) it must be behind the OTHER one you didn't choose.

2/3 is better than 1/2, so you're better to change from the choice you
make, which in the long run will win exactly 1/3 of the time, hence
lose 2/3 of the time, and always pick the one of the other curtains
(jointly winners 2/3 of the time) that is NOT sure to be empty, as the
open one is, for it "inherits" the full joint 2/3 chance that that one
(or the other) of the two you had NOT chosen in the first place will
contain the prize!

An even more dramatic way of looking at it is this: Supposing the game
was called off (by the same person who knew where the prize was) every
time your first choice was correct (i.e., it's called off 1/3 of the
time -- exactly the 1/3 when you would have won)! That would mean that
whenever the game was NOT called off, and he showed you an empty
curtain, you would KNOW that it had to be under the last curtain (and
you would, I hope, unhesitantly choose that one, rather than sticking
to your first choice!). How often would you win with these new rules?
1/2 the time or 2/3 of the time? It should be obvious that it would be
2/3, since the game would be called off 1/3 of the time, and all the
other times you would win! Well, if you put your own first choice back
into play, the situation is pretty much the same: You renounce the 1/3
of the time your first choice would have won for the 2/3 of the time
the other one will win.

Now, as SOON as you get it, run off and explain it to 2-3 people; that
way you will hammer in the insight for a lifetime...

Chrs, Stevan

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