Many thanks to Freddie for taking on Palmer. I’ve found it entertaining. I shall attempt to clarify: Palmer is citing a very specific (and, I agree with Freddie, rather obscure and all but irrelevant) variation in which Monty’s criteria for opening a “goat” door are both non-random and known to the contestant. In Palmer’s scenario, Monty has a preferred door, and his only reason for not opening his preferred door is that it will reveal the prize. If Monty fails to open his preferred door, the contestant gains information about where the prize is (and it’s not behind his own door), making his odds of success 100% by switching and leaving the cases where Monty opens his preferred door to 50/50. In either case, Monty’s opening of the door gives the contestant more information because the contestant knows Monty’s non-random strategy. The original calculation of “it’s always better to switch” is based on the premise that Monty’s action of opening a door does not change the 1/3 odds on the contestant’s originally chosen door because the contestant gains no information about his original choice.

]]>… plus 0/3 (chose Monty’s open goat door). ]]>

Exactly. As observers, we do not know what strategies the contestant or host may be using (although we know they are not colluding, as any suggestion of that is omitted from the rules and therefore not material to them).

This means that as observers, we do not know which scenario has led to Monty opening Door2. Being unable to separate out the individual occasions when it might be 50/50, we are left looking at a series in which it is 1/3 (stick) plus 2/3 (switch) to win the car. ]]>

In many (if not most) statements of the MHP Monty’s strategy is not defined.

The most reasonable assumption to make, and the one practically everybody makes without even realising it, is that he picks a door at random – which gives 1/3 and 2/3 for each game.(Oddly though most people don’t apply this same reasoning to the “I have 2 children at least one is a boy ….” problem).

It’s not however the only interpretation of the problem that can be made (as the blog writer has noted in his opening comment)

But Monty doesn’t do that in the MHP. ]]>

On average it’s 1/3 and 2/3, but in an individual game it isn’t. I’m not wrong.

Your statement “where Monty does not know how the contestant made their first choice and cannot base his strategy upon it” is wholly irrelevant, the contestant’s 1st choice is academic, and the ONLY strategy at Monty’s disposal is which goat door to open when he has a choice of 2

]]>Your scenario will occur one in six (I think) times the genuine Monty Hall game is played, but the probability of winning will remain stick = 1/3, switch 2/3 not 50/50. The probability will not change just on those occasions the scenario occurs.

You have described a new game where the contestant has a strategy which Monty knows. This does not have any bearing on the probability in the genuine MHP (where Monty does not know how how the contestant made their first choice and cannot base his strategy upon it).

Raises an interesting slant though – did Monty use his charisma to influence their final choice in a way that would increase dramatic tension and so improve ratings? ]]>