Re: [Phys-l] Monty Hall problem

[John Mallinckrodt <ajm@csupomona.edu>]

John Denker wrote:

> Game 5:  Monty is your adversary, and is free to pursue any
>   strategy he likes.  He gets to choose whether or not to
>   make the offer.  His best strategy is: _________________
> Your strategy (5):  _____________________________
> Result: ________________________
>
> Again, fill in the blanks.  What is his best strategy?
> What is your best strategy?  What is the result if each
> of you pursue your best strategy?


Making the (still unrealistic) assumption that there are no other  
subjective clues for Monty to use in making his decision to "show a  
goat and offer a switch" or for you to use in making your decision to  
"accept the switch," the probability of winning the car for any given  
strategy can be expressed in terms of the probabilities

C (you pick the car, Monty offers the switch)
G (you pick a goat, Monty offers the switch)
S (you exercise the offer to switch if received)

I find that Monty's best strategy is always to offer when you pick  
the car and never to offer when you pick the goat and that your best  
strategy is never to switch.  This, of course, leads to winning the  
car 1/3 of the time.

That's not terribly surprising or interesting, but it is somewhat  
interesting to note that:

If C is less than twice G than, then the odds of winning  
monotonically increase (beyond 1/3) as S becomes greater.
If C equals twice G, then the odds of winning are 1/3 independent of  
the value of S.
If C is greater than twice G, then the odds of winning monotonically  
decrease (below 1/3) as S becomes greater.

John Mallinckrodt
Cal Poly Pomona