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SWilson
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Joined: February 13th, 2018, 5:27 pm

Card Removal Theory in Poker

February 14th, 2018, 10:35 pm

Speaking of texas hold'em here.  So the concept behind card removal in poker is that your opponent(s) are less likely to have the  two cards that you have been dealt.   In actuality its impossible for them to have the EXACT cards you have, obviously.  But lets say you have Ace, 7 as your 2 cards.  The flop comes out 4,5,6.  Well we would be worried about our opponent having 7,8  or 3,7 as those two cards would make him a straight.  But since we hold a 7, he has less combination of hands that can contain a 7 and this gives us a slight statistical edge in our favor of our opponent not making a straight.  When playing heads-up, 1 on 1, the basic math around poker holds up really well as there are only 2 given cards out of the deck in our opponents hand and we can model around that and create +/- odds in a range.

This is where it gets really interesting and I've thought about for a long time.

A full poker table is 9 or 10 players, lets say 9 for us, including ourselves.  We are dealt two cards, both of them hearts.  The dealer deals out the flop, 3 cards and 2 of them are hearts.  So now we have 4 cards to a flush with still two more cards to come, the turn and then the river.  The BASIC math on the odds of another heart coming out to make our heart flush is ROUGHLY 1/3.  It's 1/4 odds twice, but when you factor in I have 2 hearts and there are 2 hearts on the flop there are less hearts now in the deck so the odds come out to about 33% with our 2 more cards to come.  This is generally accepted odds in poker that is applied to all situations.  VERY high level here, in our heads up example above our opponent can also have two hearts which would lower the chance of another heart coming out, or he could have other non hearts which would increase our chance of making our flush by a quantifiable amount. In this case we can still go with our general 33%.  Ok fine.   BUT......In our full table example it is POSSIBLE that all other 8 players were dealt hearts which could make the chance for another heart to come out 0, they've all been dealt out and folded so no hearts left to come.  But in contrast, if all the other cards dealt were non-hearts our chance would go significantly up but never to 100% since there would still be non heart cards left in the deck.  

So in heads up play the deviation from our 33% can be measured but perhaps not enough to effect our odds enough to not apply accepted game theory.  But in a full table example those odds can be effected significantly and possibly less in our favor than more.  Does anyone have any fun thoughts around this from a risk perspective?  The theory is that the probability of dealing out hearts and non hearts would be evenly distributed as to not effect the 33% but I have trouble agreeing with that.  Would be a fun student project......... is the upside and downside risk disproportionate when more players are added.