The Monty Maul Problem

In his Cod­ing Hor­ror Blog, Jeff At­wood writes about the Monty Hall Prob­lem and some var­i­ants. The clas­sic prob­lem pre­sents the situ­a­tion in which the game show host al­lows a con­tes­tant to choose one of three doors, one of which opens to re­veal a prize while the other two re­veals goats. The host then opens one of the other doors, re­li­ably choos­ing one that has a goat, and in­vites the con­tes­tant to switch to the re­main­ing un­opened door. The prob­lem is to de­ter­mine the prob­a­bil­ity of win­ning the prize by switch­ing and stay­ing. The var­i­ants deal with cases in which the host does not re­li­ably choose a door with a goat, but hap­pens to do so.

Jeff cites Monty Hall, Monty Fall, Monty Crawl (PDF) by Jeff Rosen­thal, which ex­plains why the var­i­ants have differ­ent prob­a­bil­ities in terms of the “Pro­por­tion­al­ity Prin­ci­ple”, which the ap­pendix ac­knowl­edges to be a spe­cial case of Bayes’ The­o­rem.

One of Jeff’s anony­mous com­menters pre­sented the Monty Maul Prob­lem:

Hy­po­thet­i­cal Si­tu­a­tion:

The Monty Maul prob­lem. There are 1 mil­lion doors. You pick one, and the shows host goes on a bloodrage fueled binge of in­sane vi­o­lence, knock­ing open doors at ran­dom with no knowl­edge of which door has the car. He knocks open 999,998 doors, leav­ing your door and one un­opened door. None of the opened doors con­tains the car.

Are your odds of win­ning if you switch still 5050, as out­lined by the linked Rosen­thal pa­per? It seems counter-in­tu­itive even for peo­ple who’ve wrapped their head around the origi­nal prob­lem.

If you take as ab­solute the prob­lem’s state­ment the host is ran­domly knock­ing doors open, then yes, the fact that only goats were re­vealed is strong ev­i­dence that only goats were available be­cause you picked the door with the prize, which, when com­bined with the low prior prob­a­bil­ity that you picked the door with the prize, gives equal prob­a­bil­ity to ei­ther of the un­opened doors hav­ing the prize.

How­ever, the fact that only goats were re­vealed is also strong ev­i­dence that the host de­liber­ately avoided open­ing the door with the prize, and there­for switch­ing is a win­ning strat­egy. After all, the prob­a­bil­ity of this hap­pen­ing if the host re­ally is choos­ing doors ran­domly is 2 in a mil­lion, but it is guaran­teed if the host de­liber­ately opened only doors with goats.

Note that this prin­ci­pal still ap­plies in var­i­ants with fewer doors. Un­less there is an ac­tual penalty for switch­ing doors (which could hap­pen if the host only some­times offers the op­por­tu­nity to switch, and is more likely to do so when the con­tes­tant chooses the win­ning door), any un­cer­tainty about the host choos­ing doors ran­domly im­plies that it is a good strat­egy to switch.