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# The Donny Hall puzzle

## Question

Monty Hall was out sick today, so his brother Donny Hall filled in as the host of Let's Make a Deal. But people quickly discovered why Monty was chosen as the regular host in the first place. It was a bad sign when Donny asked a contestant "Would you like to keep that \$100, or trade it for the trip to Hawaii inside the mystery box?" Worse was when he offered prizes for drawing a heart or spade and zonks for drawing a diamond or club, but presented the deck of cards face-up for contestants to draw from. Although the producers are sobbing with their heads in their hands off in the production booth, they haven't yet had a commercial break to give young Donny a crash course in hostmanship. And now he comes to you with an offer to win a car, a donkey, or a lifetime supply of air behind doors #1, #2, or #3. You choose door #2. Donny continues with "All right, the first door you didn't pick was door #1 so let's see what's behind that door." It's opened to reveal a lifetime supply of air.

Now you're given the chance to stay with door #2 or switch to door #3. (Donny also offered the chance to switch to door #1, but let's ignore that option for the purpose of this puzzle.) Which door should you pick?

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Assuming "first" means lowest-numbered,

Spoiler

I'm thinking it doesn't matter.

So long as the choice of the opened door is not based on knowledge of what's behind it (so for example Donny might have shown the car), and so long as the car placement was random, and not biased with respect to low numbers, we can consider that anything can be behind any door with equal likelihood.

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Going by previous history the door Donny tells you the car is behind.

Actually thinking about it no matter which door you picked Donny would never show you what is behind door number 3.  Going by previous history (his lack of hostmanship) this could be an indication that the car is behind door number 3.  If you picked 1 or 2 first then you should swap.

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If you can get away with telling Donny "I pick the door with the car" then that's better than taking any chances.

If that fails, I tend to agree with bonanova.

Even if Donny ends up doing the exact same thing that Monty does in the original Monty Hall puzzle, the answer is completely different.

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Around World Series time recently, (that's the American baseball championships for our international Denizens) I asked the question, dressed as a probability puzzle, "Given that you could pick one of the best-of-seven games in the series that you could guaranty to win, which game would that be? I even flavored the question with the opinions of experts that the pivotal games are often taken to be game 3 or game 5, as statistically these games correlated well with the eventual winner. And many responded with one of those choices for their answers. But the correct answer of course is game 7. (Or as some said, the Last game.)

So I also like the "one with the car" choice.

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If Donny could have shown the car, he would have... Unless he really thinks the lowest numbered door should always be offered instead. In that case, it is 50/50.

On the other hand, if you have already chosen the car, why would Donny even bother with another door? Maybe this is Donny trying to help you out? Thus, switch.

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