Re: River card syndrom (cross-posted in psychology)
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I'm revisiting this old thread (from the psychology forum), to share some new thoughts in the subject. (The subject was why people tend to see the river card as having more "powers" than any other board card in HE, and why they feel they're often get outdrawn on it. There were some very interesting replies on the psychology forum, specifically by PDosterM)
Assuming player A holds AA, and player B holds 22. They are both all-in pre-flop. 22 will outdraw AA a little less than 20% of the time.
I'm doing here a really rough calculation, *without redraws*, not to complicate it. So, I assume 22 beat AA by hitting his set card.
Now, looking backwards, on what card of the board was it more probable that player B would hit his set? (I'm calculating the outs from a 48-cards-deck to begin with, since we are accounted for 4 known cards)
First card: 2/48 = ~0.04167
Second card: 2/47 = ~0.04255
Third card: 2/46 = ~0.04347
Turn card: 2/45 = ~0.04444
River Card: 2/44 = ~0.04545
It is clear that the chances player B outdrew player A by hitting his set *on the river*, are bigger than the chances he hit it on any other specific board card. Not by much, but the difference is there. I assume this kind of small difference will be there for any draw, against any made hand.
This is a very simplistic calculation, but I believe it sheds some interesting light on this "river card syndrom".
Any thoughts? Especially regarding the mathematic side of this problem?
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The math here is off. The probability of 22 hitting a set on the 2nd card (which implies that it didn't hit the set on the 1st card) is
( (1 - p(hitting set on 1st card) ) * (2/47)), or ( (46/48) * (2/47) = about .04078
3rd card = (46/48) * (45/47) * (2/46) = .039989
4th card = (46/48) * (45/47) * (44/46) * (2/45) = .03900
River card = (46/48) * (45/47) * (44/46) * (43/45) * (2/44) = .03812
What you are really asking is: Given that one player has AA and the other has 22, what is the probability that the first of the remaining 2's comes at the 1st, 2nd... card.
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