Monty Hall Problem: Tree Diagram

[Domenico Rosa]

The Monty Hall problem has been discussed in several posts to this
list. The e-mail newsletter CHANCE News 7.04 reported that this
problem has resurfaced in The Independent (London). Two articles,
written by William Hartston on 28 Mar 1998 and 11 Apr 1998, can be
obtained over Lexis-Nexis.
I am taking the liberty of forwarding the following tree-diagram,
which I constructed in December 1990 and is based on the assumptions
made by Marilyn vos Savant. Namely:

1. The initial placement of the auto (car) is done at random.
2. The contestant chooses a door at random.
3. The host will not open the contestant's door and will not
open the door containing the auto.
4. If both remaining doors contain a goat, the host will open
one at random.

The notation denotes the following sequence of activities:

Ai = the auto is placed behind door i, for i = 1, 2, 3.
Cj = the contestant chooses door j, for j = 1, 2, 3.
Hk = the host opens door k, for k = 1, 2, 3.

Auto Contestant Host Proba-
Placed Chooses Opens Outcome bility

* H2 A1C1H2 1/18
1/2 *
*
C1
* *
1/3 * 1/2 *
* * H3 A1C1H3 1/18
*
* 1/3 1
A1 --------------> C2 ------> H3 A1C2H3 1/9
* *
* * 1
* 1/3 * C3 ------> H2 A1C3H2 1/9
*
1/3 *
* 1
* C1 ------> H3 A2C1H3 1/9
* *
* 1/3 *
* * * H1 A2C2H1 1/18
* * 1/2 *
* 1/3 * 1/3 *
O ---------> A2 -------------> C2
* * *
* * 1/2 *
* * * H3 A2C2H3 1/18
* 1/3 *
* * 1
* C3 ------> H1 A2C3H1 1/9
*
1/3 *
* 1
* 1/3 * C1 ------> H2 A3C1H2 1/9
* *
* * 1
A3 --------------> C2 ------> H1 A3C2H1 1/9
* 1/3
*
* * H1 A3C3H1 1/18
1/3 * 1/2 *
* *
C3
*
1/2 *
* H2 A3C3H2 1/18

When this game is played under the vos Savant assumptions, there
are 12 elementary outcomes. Each outcome where the auto is behind
the contestant's door has probability 1/18. Each outcome where
the auto is behind the remaining door has probability 1/9.
It follows that the conditional probabilities are 1/3 and 2/3 of
the auto's being behind the contestant's door and behind the
remaining door, respectively.
I would like to express my gratitude to Professor Dawson Fulton
who taught me Probability and Statistics when I was a college
junior in 1968-69. Bertrand's Box Paradox, which is similar to
the above, is one of the homework problems that taught us the
concepts of conditional probability and Bayes' Formula.

Domenico Rosa

[Milo Schield]

Domenico Rosa wrote in message ...

>
>The Monty Hall problem has been discussed in several posts to this
>list. The e-mail newsletter CHANCE News 7.04 reported that this
>problem has resurfaced in The Independent (London). Two articles,
>written by William Hartston on 28 Mar 1998 and 11 Apr 1998, can be
>obtained over Lexis-Nexis.
>I am taking the liberty of forwarding the following tree-diagram,
>which I constructed in December 1990 and is based on the assumptions
>made by Marilyn vos Savant. Namely:

An excellent, comprehensive summary.

I find the tree diagram to be more tractable by assuming the prize is behind
a fixed door (say #1). The naming of doors is totally random and the person
making the choices is ignorant of the naming, so I don't see any loss of
generality.

But now the tree is considerably pruned, so that one might better see the
shift in conditionality.