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June 23rd, 2019, 01:04 PM  #1 
Newbie Joined: Jun 2019 From: Chapel Hill NC Posts: 1 Thanks: 0  Hi. New user here with a math question
First off, if you're unfamiliar with Texas hold 'em poker, let me tell you how it's played. First, each player is dealt two cards and there's a round of betting. Next, 3 cards are put on the table that belong to all the players and another round of betting, then a fourth card on the table and another round of betting. Finally, a fifth card on the table and another round of betting and the player with the best 5 card combination of the cards in his hand and the cards on the table wins. Now I have created a database with 400,000 simulated games (minus the betting. I have no idea how to simulate that). What I'm trying to determine is what is the optimum strategy for when to fold after the first two cards are dealt. So I have a table with a criteria id and 5 parameters, I apply these parameters to the simulated games to determine whether or not that person folded and whether or not they ended up winning the game. There are two ways to measure success. How many times did I fold when I shouldn't have? You want that number, let's call it A, to be as low as possible. Or How many times did I do the right thing by folding? You want that number, B, to be as high as possible. What I found is that strategies that scored very low on A also score very low on B and strategies that score high on B also score high on A. So the optimum would be measured by some formula using A and B. I tried B  A and that seemed to favor the low A scores. I thought that since there are 3 losses for every win (average four players per game), try B*3 A which heavily favored the high B strategies. I feel in my gut that the optimum strategy is one in the middle, but I can't figure out what it is. Thoughts anyone? Last edited by skipjack; June 24th, 2019 at 12:33 PM. 
June 24th, 2019, 12:40 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,802 Thanks: 2149 
Fold with probability p, where you try to find a formula for p that depends on your cards.

June 24th, 2019, 03:20 PM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 634 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
The sample space for this problem is absolutely enormous. Like, so large it would be useless to try and comprehend. This is the kind of problem for which machine learning has been extremely successful. As you might imagine, its already been done for texas holdem adding it to the list of (rapidly growing) games which computers are better at than the best humans on earth. https://www.deepstack.ai 
June 29th, 2019, 09:57 AM  #4 
Member Joined: May 2018 From: Idaho, USA Posts: 49 Thanks: 5 
I am a Texas Holdem player also. I have thought of the same question you are asking. I have also thought of the chances of getting certain hands and knowing when to fold. There are 2,598,960 possible poker hands total according to the internet. Depending on what the two cards are in your hand, and the three cards on the flop, You know what hand you have, and what the chances are of getting a better hand on the turn and river. Here is a real life scenario from my experience on April 30th, 2019. I have a picture of it: I was dealt an Ace of clubs and a two of clubs. There were 5 other players, each with 2 cards. This means there were 40 cards left in the deck. On the flop, it was a Jack of clubs, a 3 of clubs, and a 4 of clubs. I had a flush. I then thought to myself the following: There is a chance that another player has a 5 of clubs, there is also a chance it was discarded when burning a card for the flop, and there is a chance it is not in a position in the deck to show itself. A low chance in the end of getting a straight flush. I did not fold, because I had a flush and the odds of winning were good so far. On the turn, a 5 of clubs. I had a straight flush. Now, lets look at this from the point of view of the opponent who called me and lost. He had a pair of 2s in his hand, specifically the spade and diamond. He might have thought this on the flop: He may have a flush, but the odds of that are low. On the turn, if I get a 2, Ill have three of a kind and may beat him. When playing Texas Holdem, you kinda have to do the math in your head, and figure out what the odds are that your opponent has better cards, the card you need to get a better hand is available on the flop or river, and so on. What I do is if the odds are very low, I try not to take a chance and fold. If they are high, like when I got the straight flush on the turn, I take a chance. When you know all the odds and factors, that is when you decide if you want to fold or not, and it all depends on the odds. To quote Kenny Rogers, "You got to know when to hold them, know when to fold them, know when to walk away, and know when to run." As @SDK said, The numbers are absolutely enormous. Finding them out would be fun, though! Good luck, Jared 

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