Card Sharks

[AaronBrown]

[ QUOTE ]
First, I think you switched the labels on these cases.

Second, I get 4*(27+31+35+39+43+47)=888 ways to win twice and 4*(23+19+15+11+7+3)=312 ways to win and then lose. As should be expected, these total 888+312 = 1200 = 6*4*50, since there are 6 ranks that win for you, 4 cards in each of those ranks, and 50 possible third cards given the first two.

Did you assume aces are low?

[/ QUOTE ]

Whoops! You're right. But my post sounded good. I used the same method, but somehow moved the nines on the second card from winners to losers. That's why I'm 4*27 = 108 short on the WW and 4*23 = 92 short on the WL; with those cases moved to LW and LL respectively.

You're also right that I mixed up the labels, of course to win twice after a seven you need a nine or higher on the second card, and the third card to be lower than the second card, not higher.

No, I assumed Aces were high.

In an attempt to salvage some credibility, here is a simple example of the principle I was defending. There is an urn with one white marble and two black ones. You have $1,000 and are allowed to bet on two draws from the urn. There is no replacement, once a ball is drawn it is discarded. The house pays $1.80 for each $1.00 bet if you win.

On your first bet you have 2 chances in 3 of winning if you bet on black. (2*$0.80 - 1*$1)/3 = $0.20 so this is a positive expected value bet. However, if you win the first bet the odds are even on the second bet, so you have a negative expectation and wouldn't bet. But if you lose the first bet you're guaranteed to win the second, so you would bet everything.

If you bet X on the first draw and lose, you end up with 1.8*(1,000 - X). If you win, you end up with 1,000 + 0.8*X. The first happens one time in three, for an expected value of 0.6*(1,000 - X) and the second happens two times out of three for an expected value of (2,000 + 1.6*X)/3. Add them together and you get (3,800 - 0.2*X)/3. So you should bet nothing on the first draw, despite the positive expected value.

[einbert]

Wow, awesome thread. I have really enjoyed reading this and thinking about how these concepts might apply to poker. I particularly enjoyed the marble story, I'm definitely going to have to put even more thought into that kind of situation, very interesting.

You guys seem like you might be interested in this hypothetical MTT situation that I just posted, about an extremely high variance decision made very early in a large field MTT. I would really be interested in what you guys have to add to the thread, if you feel like checking it out.

[AaronBrown]

That was a good question.