I still can't see how the odds for the second choice could be anything but
50/50.  Sure, Monty deliberately avoided the door with the WBDOA.  But that door
could well be the one you've already chosen.  In fact, as I see it, the odds
were really 50/50 even before the first choice was made.  Since we know in
advance that Monty is going to open one of the doors, and we also know that he
will always choose a goat, then there was never really a choice of three doors.
In effect, the first choice doesn't count, because whatever choice is made will
eliminate one of the goats.  So effectively, the real choice is always going to
be between the other two doors, one of which hides the WBDOA, and there'll be a
1 in 2 chance of choosing it.

Wayne





Bruce Toback <[log in to unmask]> on 03/01/2000 12:54:33 PM

Please respond to Bruce Toback <[log in to unmask]>

To:   [log in to unmask]
cc:    (bcc: Wayne Brown/Corporate/Altec)

Subject:  Re: OT: Probability question



Gary Paveza writes:

>Every time I hear this one I scratch my head in confusion.
>
>At the beginning, you have a 1 in 3 chance of guessing the right door.

Here's a copy of the response I wrote last time (see
<http://www.optc.com/chemed-l-thread/Thread1980.html> for the full
discussion):

>I probably shouldn't answer this, but it's more fun than what I *should*
>be doing. :-)
>
>When you first choose one of the three doors, there's a 1/3 chance that
>you're standing on the wealth beyond dreams of avarice (WBDOA). Most
>people focus on this, which is the source of the difficulty. But there's
>a 2/3 chance that you're standing on a goat, and it's easier to
>understand the reasoning from that perspective.
>
>When Monty shows you a goat (remember, he will *never* show the WBDOA),
>that original 2/3 doesn't change, because opening the curtain doesn't
>change the state of your knowledge before you chose: there's still a 2/3
>chance that you're standing on a goat. But since Monty has now shown you
>a goat, that means that there's now a 2/3 chance that the door you didn't
>pick is the one with the WBDOA.
>
>It's easy to think that when Monty exposes the goat, the chances are now
>50/50. The reason it's not that way is that Monty's choice wasn't random:
>he deliberately avoided the door with the WBDOA. If Monty had opened a
>random door and exposed a goat (which would happen half the time), then
>it really wouldn't matter whether you switched.

This is another case where it's easiest to see the reasoning if you list
all the possibilities.

-- Bruce



--------------------------------------------------------------------------
Bruce Toback    Tel: (602) 996-8601| My candle burns at both ends;
OPT, Inc.            (800) 858-4507| It will not last the night;
11801 N. Tatum Blvd. Ste. 142      | But ah, my foes, and oh, my friends -
Phoenix AZ 85028                   | It gives a lovely light.
btoback AT optc.com                |     -- Edna St. Vincent Millay
Mail sent to [log in to unmask] will be inspected for a
fee of US$250. Mailing to said address constitutes agreement to
pay, including collection costs.