Bear with me through another variation of the Monty Hall problem, which I began a week or so ago.
Recall that I contended that it was crucial (or so it seemed) that Monty knew where the prize had been placed. But try this variation. The rules are the same: there are three doors, one of which hides $60,000 in cash; the other two, goats. You've been selected to play the game, and you choose Door 3. Before anything else can happen, one of the doors at the back of the auditorium opens, a breeze rushes through the auditorium, and Door 1 is blown open to reveal a goat. (It was chance, not the knowing agency of Monty that revealed the information regarding Door 1's contents to you.)
Monty is unexpectedly faced with a situation not of his own devising. Nonetheless, he offers you the same deal as before -- keep Door 3, switch to Door 2, or take $35,000 in cash (on the assumption that you are risk neutral). What to do?
Here's my analysis, but I'm not convinced that it's correct. I think that you should still switch. The information that has been revealed still fits the model developed before -- namely, that there was a two-thirds probability that the prize lay behind Doors 1 and 2. (Remember that the prize has been placed behind some door and stays there, unlike the situation in the black velvet back variant.)
But there's a subtle point here that bears further thought. If we were to incorporate the door-opening breeze possibility into our scenarios, then things get more complicated. One-third of the time the breeze would blow open the door hiding the cash because the breeze, unlike Monty, doesn't know where the prize has been placed. And one-third of the time the breeze will blow open the door that you originally chose. So, to be as clear as I can, in the door-opening breeze scenario, you should switch only if the breeze has blown open one of the two doors that you did not choose and that does not contain the cash prize. (In only that circumstance the breeze mimics Monty's knowing opening of the door hiding a goat.)
But I may be wrong. Please correct me if I cam.
I may do one last posting on Monty, incorporating the comments of my friend Charlie Petit. Till then, look at this Google page for references on Monty. And try it with others using three cards, placed face-down, one of them designated the winner.
TSU
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