There were no 0 zero proposed answers to the Monty Redux problem.

I give my answer below. New bits are in ** old bits are in "".

Recall

"Monty Redux

Recall, the Monty Hall paradox from “Let’s Make a Deal” an old game show. The last step in the game show was guess which of 3 boxes contains a big prize. There are three large boxes, box 1, box 2 and box 3, one contains a very valuable prize, the other two contain prizes of small value. The contestant guessed a box. The assistant (Carol Marol or was she on truth and consequences) opened one of the other two boxes showing that it didn’t contain the big prize. Monte Hall offered the contestant the chance to switch to pick the remaining unopened unselected box.

Now all contestants must have noticed the pattern that the assistant always opened one of the boxes which had not been chosen and that box always contained a small prize. "

*This, I think is the, often unstated, assumption that makes the problem such a dread dinner party troll. If there is no information that a box with a small prize is always opened, there is no way to solve the problem. It might be that a box is opened only if the contestant guessed right the first time (so better to not switch and win for sure ) or only if the contestant guessed wrong the first time (so better to switch and win for sure) or anything in between. It was clear on the show that a box was always always opened.*


"I am going to add another assumption to make the assistant follow a well defined rule. I assume that the assistant opens the box which 1) has not been chosen by the contestant 2) does not contain the big prize and 3) has the lowest number of boxes satisfying 1 and 2.

Now consider a case of the game. Contestant guesses box 2. The assistant opens box 1 showing that it contains a small prize ? The contestant is allowed to switch and guess that the prize is in box 3. Should the contestant switch ? Does it make any difference for the probability of winning ?"

*In this the chance of winning is the same (1/2) whether the contestant switches or not. Given rule 3 stated above, box 1 is opened whether the big prize is in 2 or in 3 so no information is revealed.*



"How about another case of the game. Contestant guesses box 2. The assistant opens box 3 showing that it contains a small prize ? The contestant is allowed to switch and guess that the prize is in box 1. Should the contestant switch ? Does it make any difference for the probability of winning ? "

*here given rule 3, opening box 3 means the big prize must be in box one so the contestant should switch. The point (if any) of the new puzzle is that it the other case which shows that it can sometimes be costless to switch (a knife edge result).*
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