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Server Time: 12/1/2008 9:24:38 PM PACIFIC |
 Will you flop a pair as often as you think?, Chris W, 19. Dec 2003 17:57 | ||
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| Everyone that has studied the basic odds behind hold'em knows that the probability of flopping a pair when starting with 2 unpaired hole cards is 32.43%. This assumes all 6 of the cards you need are live (a player who folded didn't flip up a card,the dealer didn't flash one, etc.). However, this assumes that you know nothing about the distribution of the remaining 50 cards and this may not always be the case. The actions of the other players can often give you some information that leads you to believe that one or more of your cards may not be live. If one of your 6 cards are out, your chances fall to 28.1%, if 2 are out its 23.4%. As an example, assume you raise with AK from EP and get 6 callers. It's almost certain that unless your opposition is extremely weak, at least one person is holding another A or K, thus reducing your chances of flopping a pair significantly. Similarly, if you limp in LP behind 3 players with two face cards like QJs its likely one of the limpers has one of your cards. So it would appear that simply by seeing the flop with multiple players entering voluntarily some of the randomness in the distribution of the 50 other cards is reduced. But how can we quantify this? Obviously its not possible to generate an equation to describe this behavior but the following factors would come into play: - # of players entering the pot - the ranking of your hole cards (higher cards have a greatly likelihood of being played) - whether or not there has been a raise - the quality of your opposition (weaker opposition gives you less info about their hands) In general, it should be harder to flop pairs with quality cards while crappy hands are more likely to flop pairs at the 32.43% rate. The main reason I have for asking this question is that I've preformed a statistical analysis of my poker tracker database that produced surprising results. I looked at all of the quality non pair hands (AK,AQ,AJ,AT,KQ,KJ,QJ,JT) that I played where I saw the flop to see how often I flopped a pair. It turns out that I only flopped a pair 23.5% of the time, or 440 times out of 1872. This is actually -8.25 standard deviations away from the mean so clearly the percentage of times that you make a pair under these conditions must be < 32.43% (unless there is a serious problem with a random number generator). I'd be very interested to know if anyone has seen any kind of statistical analysis or simulation data on this topic. Thanks, Chris | ||
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| Re: Will you flop a pair as often as you think?, Roy Cooke, 20. Dec 2003 08:37 | ||
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| Hi Chris This is a very interesting post ....You should have flopped 606 pairs or better....... That said, the 32.4% is based on 50 unseen cards as is the case if you have not seen any other cards. Yes, it is true if many players call in front of you that the propensity for an Ace to be in their hand adjusts the equation, but not by that much as you will also have to include the cards in other players hand if you wish to calculate correctly....Like if you see zero flush cards of yours in 7 stud the chance of making a flush goes UP over if you base the equation on unseen cards.... That said, I think you are mostly running bad! You are still playing big cards if nobody calls and the propensity then would go up...those factors even themselves out pretty much...... The difference you state is too big to assign to card distribution. Life is Good :-) Roy Cooke on 19. Dec 2003 17:57 Chris W wrote: > Everyone that has studied the basic odds behind hold'em knows that the > probability of flopping a pair > when starting with 2 unpaired hole cards is 32.43%. This assumes all 6 of the > cards you need are live (a player who > folded didn't flip up a card,the dealer didn't flash one, etc.). However, this > assumes that you know > nothing about the distribution of the remaining 50 cards and this may not > always be the case. The actions of the other players can often give you some > information that leads you > to believe that one or more of your cards may not be live. If one of your 6 > cards are out, your chances > fall to 28.1%, if 2 are out its 23.4%. > > As an example, assume you raise with AK from EP and get 6 callers. It's almost > certain that unless > your opposition is extremely weak, at least one person is holding another A or > K, thus reducing your > chances of flopping a pair significantly. Similarly, if you limp in LP behind 3 > players with two face > cards like QJs its likely one of the limpers has one of your cards. > > So it would appear that simply by seeing the flop with multiple players > entering voluntarily some of > the randomness in the distribution of the 50 other cards is reduced. But how > can we quantify this? > Obviously its not possible to generate an equation to describe this behavior > but the following factors > would come into play: > - # of players entering the pot > - the ranking of your hole cards (higher cards have a greatly likelihood of > being played) > - whether or not there has been a raise > - the quality of your opposition (weaker opposition gives you less info about > their hands) > > In general, it should be harder to flop pairs with quality cards while crappy > hands are more likely > to flop pairs at the 32.43% rate. > > The main reason I have for asking this question is that I've preformed a > statistical analysis of my > poker tracker database that produced surprising results. I looked at all of > the quality non pair hands (AK,AQ,AJ,AT,KQ,KJ,QJ,JT) that I played where I saw > the flop to see how often I flopped a pair. > It turns out that I only flopped a pair 23.5% of the time, or 440 times out of > 1872. This is actually -8.25 standard deviations away from the mean so clearly > the percentage of times that you make a pair > under these conditions must be < 32.43% (unless there is a serious problem with > a random number generator). > > I'd be very interested to know if anyone has seen any kind of statistical > analysis or simulation data > on this topic. > > Thanks, > Chris | ||
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| Re: Will you flop a pair as often as you think?, Chris W, 21. Dec 2003 14:15 | ||
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| Hi Roy, Thanks for your response. It didn't occur to me to consider that short-handed pots tend to indicate that few big cards were dealt to the players that folded. This is especially true for the games that I tend to target where any ace is going to be played in just about any situation. I agree that your odds of flopping a pair actually increase in this situation so I think that you're right that these situations should roughly cancel out the multi way hands. So the percentage of times you make a pair should be still be around 32.43% or maybe slightly smaller. Assuming this is true, these numbers look very problematic. Results that differ by more than six standard deviations are generally considered too improbable to bother with. For instance, there is only one chance in 1 billion that your results would be greater than 5.997934 deviations away from the mean. My results are -8.25 standard deviations away. So either I'm the unluckiest player on the planet or there is something seriously wrong with the " random " number generator at party poker. Happy holidays, Chris | ||
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| Re: Will you flop a pair as often as you think?, Mark Barnett II, 22. Dec 2003 11:12 | ||
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| i think you might have too small a sample size? assuming 2000 hours and 35 hands an hour *roughly a full time job playing B&M poker) with about 70k hands total dealt *years worth* what would your sample size look like? Rule #1 of Poker Circumstances alter cases Rule #2 NEVER forget rule #1 | ||
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| Re: Will you flop a pair as often as you think?, Chris W, 22. Dec 2003 11:55 | ||
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| That was my initial thought as well. It's true that for calculating win rates you need tens of thousands of hands to be accurate. That is the case because there are many variables that will affect your results over the short term. However, this case is very simple. Either you make a pair on the flop or you don't; this is what is called a binomial distribution in probability theory. All of the probability theory I've read suggests that a sample size as small as 200 is sufficient for this type of analysis provided that the mean is larger than five (it is 607 in this case). As an aside, the sample comes from the 33,000 hands I've played over the past three months or so. I would expect half a year's equivalent of B&M play to converge towards the mean. so I'm fairly sure that my sample size is large enough but I'd welcome any mathematical reasons why this isn't the case. | ||
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| Re: Will you flop a pair as often as you think?, JJSCOTT2, 2. Jan 2004 02:06 | ||
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| I have two questions, first, remind me how you figured out the standard deviation(its been a while since statistics class) and second, if i remember correctly for a normal distrubution anything outside 3, maybe 4 standard deviations if you want to be really accurate would be an insane outlier. So -8.25 it seems to me would just have to be an insane run of bad luck. | ||
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| Re: Will you flop a pair as often as you think?, Chris W, 2. Jan 2004 02:44 | ||
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| anything more than six deviations is generally considered to be outside the believable range. So yes, my results are very bizarre. If you got these type of results when rolling dice you would have to conclude that the dice are weighted. here are the formulas you use to calculate this stuff. z = (k-u)/s Is the number of deviations from the mean where: u = np [the mean of the binomial sampling distribution] s = sqrt[npq] [the standard deviation of the binomial sampling distribution] n = the number of opportunities for event x to occur; k = the observed or stipulated number of occurences of event x; p = the probability that event x will occur on any particular occasion; and q = the complementary probability that event x will not occur on any particular occasion there are also applets on the web that will calculate this stuff for you if you like. Chris | ||
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| Re: Will you flop a pair as often as you think?, Formless, 20. Dec 2003 10:06 | ||
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| Good Post. I was wondering about this myself. >But how > can we quantify this? > Obviously its not possible to generate an equation to describe this behavior I think it is possible to generate an equation to describe this, but it's not the type of thing you do on the back of a cocktail napkin. Like if I have KJ on the button and there are two limpers, what are the odds they or the estimated callers behind me have a K or a J? You can always estimate a range of hands the limpers would play and the percentage of time they would play them (visualized like a 3D matrix), it's just a question of how accurate your estimate is. You need some pretty sophisticated software to do this unless your name's Stu Ungar or Raymond Babbit. But once you experiement and analyze a bit, you can get a feel for the numbers and calculate this stuff on the fly. That's my theory. | ||
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| Re: Will you flop a pair as often as you think?, PairTheBoard, 22. Dec 2003 12:00 | ||
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| it would be interesting to run some kind of simulation on this. But I suspect the Effect you're talking about is mitigated by other factors. Lots of callers may be due to play of suited connectors as well. And those who fold skew the Effect in the opposite direction. Also, I think this is akin to the theory that if everybody folds it implies a remaining deck richer in high cards and thus gives a higher likelihood that someone in the blinds will have a Big Hand. I recently read a post by Mason Malmuth on the 2+2 boards where he talked about this and claimed that considerable simulation work has been done on it, with the conclusion that THIS Effect is negligable. | ||
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| Re: Will you flop a pair as often as you think?, Chris W, 22. Dec 2003 14:47 | ||
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| I think you're probably right about this. Do you remember what thread on the 2+2 board covers this topic? I couldn't find it... Thanks, Chris | ||
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| Re: Will you flop a pair as often as you think?, Palinya, 22. Dec 2003 12:01 | ||
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| I looked at my AK and AQ in poker tracker after about 40k hands. I'm not sure how often I flopped a pair with these but I won the hand 43% and 41% of the time. This is higher than the times you'd flop a pair but I almost always raise pre-flop with these and if they miss and the board is rags, I'll bet again to represent a pocket pair. This and flopping straight / flush draws probably account for why I win more often with these than I actually hit the flop | ||
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| Re: Will you flop a pair as often as you think?, stdioh, 26. Dec 2003 20:42 | ||
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| Indeed, if you hold AK and raise it UTG, but are still called by 6 (lets say less than top shelf) players, chances are very good that a lot of your outs are used up ... of course a lot of "not your outs" are used up too. Sure your AK is more live if you have one caller than if you have 6, but not a huge amount more live. Now if the betting were to be 3-bet and then 4-bet after your raise, you can be genuinely afraid that a single player is holding 2 of your outs alone. Nevertheless, this is all moot, since when you have a lot of callers you have more odds to draw to one pair (though less chance that it'll be good if you get there) and with fewer players you have worse odds to draw to one pair with a greater chance that it'll be good when you get there. With all of this noise in the signal, wondering how live your draw to one pair is isn't all that important anymore. | ||
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| Re: Will you flop a pair as often as you think?, Chris W, 2. Jan 2004 02:50 | ||
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| I do agree with this, however I was more interested in these results from a statistical results standpoint. It just seems like something is strange with my results ... | ||
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| Re: Will you flop a pair as often as you think?, jajo, 12. Jan 2004 14:01 | ||
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| "Everyone that has studied the basic odds behind hold'em knows that the probability of flopping a pair when starting with 2 unpaired hole cards is 32.43%." Actually 32.43 is for a pair OR BETTER. The chances of flopping one pair specifically is 28.96%. See http://pokerstrategy.org/poker_stats/poker_stats_he.htm#flop_pair | ||
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