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#1
09-17-2007, 08:28 PM
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3 card Monty, and the incorrect poker math
Three cards; one ace, two blank cards. You pick one keeping it face down. After that I pick a blank card (randomly or not) and place it face up, and the other I place face down. You can now take the first or the second face down card. You win if you get the ace.
You should pick the second face down card as the odds of it being an ace are 1/2 as it was dealt after the blank card. The odds of the first face down card being an ace are 1/3 as it was dealt before the face up card. If this would be a poker game and the same question would be answered, it would be answered incorrectly. The only difference is that both down cards would have been dealt before the blank up card. That's why that "blank" up card has no effect to the preflop probabilities - it would only effect it if it would be an ace as there are no aces left. 
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#2
09-17-2007, 10:12 PM
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Re: 3 card Monty, and the incorrect poker math
If I picked the ace to begin with (33.3%) the one you don't place face up will have a 0% chance of being the ace.
If I don't pick the ace to begin with (66.7%), the one you don't place face up will have a 100% chance of being the ace. looks to me like the second face down card has a 66.7% chance of being the ace, not 50%. looks to me like you answered it incorrectly even when it wasn't a poker game.
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#3
09-17-2007, 11:05 PM
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Re: 3 card Monty, and the incorrect poker math
[ QUOTE ]
Three cards; one ace, two blank cards. You pick one keeping it face down. After that I pick a blank card (randomly or not) and place it face up, and the other I place face down. You can now take the first or the second face down card. You win if you get the ace. [/ QUOTE ] I'm concerned about the "randomly or not" part. If I don't choose the ace, what is the probability that you'll turn the ace face up? 0? [ QUOTE ] You should pick the second face down card as the odds of it being an ace are 1/2 as it was dealt after the blank card. The odds of the first face down card being an ace are 1/3 as it was dealt before the face up card. [/ QUOTE ] If you turn a blank face up one of the two remaining cards must be an ace. So the probabilty of the first being an ace plus the probability of the last card being an ace must equal exactly one. You have it at 5/6. It's 2/3 and 1/3 for Monte Hall, not 1/2 and 1/3. [ QUOTE ] If this would be a poker game and the same question would be answered, it would be answered incorrectly. The only difference is that both down cards would have been dealt before the blank up card. That's why that "blank" up card has no effect to the preflop probabilities - it would only effect it if it would be an ace as there are no aces left. [/ QUOTE ] The flop does alter your opponent's range.
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#4
09-18-2007, 12:35 AM
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Re: 3 card Monty, and the incorrect poker math
[ QUOTE ]
So the probabilty of the first being an ace plus the probability of the last card being an ace must equal exactly one. You have it at 5/6. [/ QUOTE ] I forgot to add the under 1/2 stuff of the first card.
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#5
09-18-2007, 12:53 AM
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Re: 3 card Monty, and the incorrect poker math
The Monty's 3rd card (2/3 of being an ace) is clear to me, though it doesn't necessarily drop the odds of the first card to 1/3 as that's what it already was originally. Though if the upcard is an ace (random) it drops everything. If the upcard is or isn't random, there's again something to think about.
My poker example lacked cards; both down cards could be blank cards. The point however was that if I TAKE a "blank" card from the deck after the down cards (preflop) are dealt and place it up it gives us about no more information than what we had before it (the down cards will not include that specific card, that's all). The only difference is the non-randomness in that case, and the fun part is to understand (without formulas) why it makes a difference in this case if the card comes out randomly vs. when it doesn't come out randomly.
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#6
09-18-2007, 01:09 AM
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Re: 3 card Monty, and the incorrect poker math
[ QUOTE ]
The point however was that if I TAKE a "blank" card from the deck after the down cards (preflop) are dealt and place it up it gives us no more information than what we had before it. The only difference is the non-randomness in that case, and the fun part is to understand (without formulas) why it makes a difference in this case if the card comes out randomly vs. when it doesn't come out randomly. [/ QUOTE ] I'm missing the connection this has to poker. In poker the cards are random so what comes out does alter the density function that describes your opponents' hands. In your example, taking a blank card out randomly would give you information about what your opponent held.
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