Heads Up Hold'em question

[PseudoPserious]

It's Thursday night. You are at home playing a heads-up Hold'em tournament against a friend. Your friend just dealt an unfavorable turn card, and you muck your hand face down in the burn pile after his pot-sized bet.

However, before your friend can discard his hand, the power goes out, placing the house in complete darkness.

You hear your friend fiddling with the cards. He tells you that he has picked up all 52 cards, leaving the board and his hand face-up in the deck with all of the other cards face-down as normal, and shuffled the deck.

He hands you the deck and offers you an odd bet: that you can't separate all 52 cards into two piles that both have the same number of face-up cards before the power comes back on. He generously lays you 2-1 odds.

Keeping in mind that you can not tell by sight or touch which cards are face-up or face-down in the deck, should you take the bet?

PP

[PseudoPserious]

For completeness, I guess I should have said that your friend offers you the deck in the normal manner, so that the majority of the cards are face-down.

PP

[irchans]

I say no, but it is very close. (Assuming I can split the shuffled cards nearly 26/26 and there are 6 face up cards.)

[Jimbo]

I may be incorrect but it seems like he is giving you 2 to 1 on an even money proposition. What am I missing here? I would take the bet and rather than cut the cards in half I would deal every other card into two piles. I would also go to my circuit breaker box and turn off the main switch so the power/time qualifier was eliminated.

Jimbo

[PseudoPserious]

Hey Jimbo,

I think you're missing that both your friend's hand and the board were face-up in the deck. If there were only two face-up cards, then randomly dealing out two piles of 26 would be a skosh better than a 50/50 bet. But, here we have more than 2...

That's a good idea about the circuit breaker, but I wouldn't leave the table with that particular friend sitting there. He'd be likely to take all your chips and run home. Maybe 'friend' is an overstatement. I don't even know why you play poker with him.

PP

It seems that his odds don't make a difference... I believe that if you know that there are X face up cards and 52-X face down cards, you simply need to take the first X cards and flip them upside down. That is Pile #1. The remaining is Pile #2. This way, you've got the same face up in both piles...

Example: Say there are 6 up cards... Now, say the first 6 cards are D D D U U D. By flipping you've got U U U D D U... this is 4 Up in this pile... this MUST equal the 4 up that was remaining in the other pile. This should work always.

So I'd try to get more odds out of him, but he could tell me to lay 100-1, and I'd probably take it assuming I was pretty sure the lights wouldn't go back on really quickly.

Is this what you had in mind?

[lorinda]

If I take an infinite deck of cards, and seperate them, then it will be a 3-3 split 31.25% of the time:

0-6, 6-0: 1.56%
1-5, 5-1: 9.38%
2-4, 4-2: 23.44%

Which would not be enough to take the bet, I am tired and cannot think how to approach the fact there are only 52 cards, this would certainly push you closer to the 33.334% requirement, but its gonna be tight.

[lorinda]

Nice answer, I looked for something like this and missed it, doh.

Im now fascinated about the other problem we have elected to solve (ie, if we cant flip) though your answer is evidentally the correct one.

[PseudoPserious]

Straight-up answer:

If you deal 26 cards, the odds of dealing exactly 3 of the 6 face-up are

C(6,3)*C(46,23)/C(52,26) = .33205 or so

for an EV of -.0039ish at 2-1 odds.


Tricky answer:

There are 6 upcards in the deck. Deal yourself 6 cards. Of these cards, x are face-up, leaving 6-x face-up in the rest of the deck. Flip over the 6 cards you dealt yourself. Now you've got 6-x face-up cards in each pile, for an EV of lots.


Somebody once told me to think about possible poker situations before you get to the table, so you can make the right play without having to recalculate everything everytime. However, I'm not sure this particular situation quite qualifies -- even though this "move" will guarantee you a pot, it probably not won't show up more than 2 or 3 times over your entire poker life.

PP

[PseudoPserious]

Kudos!

If no-one saw this, I was going to re-ask the question with a slightly different slant -- but you saved me the trouble [img]/forums/images/icons/smile.gif[/img]

PP