X-Message-Number: 7067
Date: Thu, 24 Oct 1996 11:18:28 -0400 (EDT)
From: "Perry E. Metzger" <>
Subject: The Monty Hall Paradox

The Monty Hall Paradox is a real paradox.

Consider:

You can ALWAYS find the correct probability by enumerating all
possibilities.

Monty is guaranteed to reveal a goat after you make your initial
selection.

One time out of every three, you will have picked the door with the
car. Monty then picks one of the other two doors at random. One time
out of three, the door you haven't chosen has a goat behind it.

Two times out of three, you will have picked a goat, and Monty will
reveal the other goat. Two times out of three, the door you haven't
picked has a car behind it.

>From: Peter Merel <>
>Subject: The paradox of unexpected goats.
> 
> Art Quaife writes,
> 
> >But this is WRONG. Your initial guess of door #1 had 1/3 chance
> >of winning. Monty's opening another door to reveal a goat, as he
> >previously promised to do, does not change that probability. But
> >since Monty has now eliminated door #3 as a possibility, door #2
> >must now have a 2/3 chance of being correct. You should switch,
> >and by doing so you *double* your chance of winning the car.
> 
> Sorry, probability theory doesn't work like this; an elementary property
> of probability theory is that past results do nothing to fix the
> odds of new measurements.

You've made a mistake, Mr. Merel.

A fundamental assumption you are using is the assumption of
independence -- that is, that you are dealing with independent
events.

> If I toss a coin and get a head, that tells me nothing about a new
> toss.

Correct. That is because coin tosses are INDEPENDENT random
variables. You are not dealing with an INDEPENDENT set of events in
the Monty Hall Paradox, however.

> The probability of the new coin toss, and the probability of the
> contestant's second pick, must be calculated based purely on the
> immediate circumstances of that event, not on its history.

Correct in the first case, wrong in the second. Monty Hall's selection
of the door is not independent of your first random choice -- it is
dependent on your selection, and reveals information, since Monty Hall
will not pick a door at random, but will only pick a door with a goat
behind it.

> So the odds of the second pick are indeed 50/50.

Nope. Sorry.

If you don't believe me, simulate this with a friend. Take three
cards, write "goat" on two of them and "car" on one, turn them face
down, and play the game about ten or fifteen times. You will see that
switching indeed gets you a probability of 2/3rds of getting the car,
and 1/3rd of getting a goat.

Perry


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