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As Harvey put it in the Discover article: Oh, and they fixed the deals, the better to measure the effects of luck. Their conclusion, per Neuroskeptic: Jack suited Ace-7 unsuited Pair of sixes. The many mathematicians and physicists who are aficionados of poker would agree with Heeb. Send a private message to dejacob.
Is there any reason why you're reluctant to use an Equity calculator for something like this? Since counting all the combos or calculating your winning chances invididually is so confusing, I'd just put the hand vs range and board into something like Equilab and get this: If this is the case then you are fine but in other situations there may be other strategic options that you would also need to calculate the EV of.
You can convert this into relative frequencies by adding up all the combos and dividing each number by the total. You can just multiple the percentages by the frequencies above. EV is the probability of an event occurring multiplied by the payout when the event occurs.
I find it easier to break out the events and then make sure I correctly identify the probabilities and the payout for each event and then adding the seperate events together to get the total EV for a particular line. Originally Posted by whosnext. Here you are deciding between folding and calling. Perhaps you implicitly are calculating the EV of calling relative to that of folding, but care must be taken here. This is equivalent to "setting" the EV of folding equal to zero. The reason I even brought this up is that this issue can be a source of confusion and errors in some EV analyses.
I don't think all of your combo counts are correct. Once you have the correct combo counts, you weight each of the respective EV's of that hand combo to derive the overall EV. The idea, of course, is to give more weight to the EV of the hand combo that villain is more likely to have and less weight to the EV of the hand combo that villain is less likely to have.
The idea is that the EV is for the different options you can take at a specific point in the hand. I think you are essentially taking the action back a step before villain shoves. But what decision are you trying to analyze? Your decision to bet the ? If so, that is an entirely different situation. It can be done, of course. In addition to taking into account the issues raised above, you must take care in handing pot amounts, including the amount of money already in the pot at the time of your decision.
Originally Posted by ArtyMcFly. So pot size at our decision point: Now at this point you could calculate the EV using the individual hands, but to make things faster we can lump hands with identical equity together just like you did previously: You can either calculate this with outs or the much better solution is to use an equity calculator since it will not miss any outs like you might.
BB code is On. Photo by Todd Klassy. Sean posed this question on Cosmic Variance back in Which hand is most likely to win if you choose to stay in the pot all the way to the showdown, against other pairs of randomly chosen hole cards? Mathematically, it depends on the number of opponents. The probability that you will win goes down as the number of opponents goes up, because there are more ways for you to be beaten. Some hands play well against very few opponents, while others play well against many opponents.
It all depends on the circumstances. Against one opponent, the sixes will win Against four opponents, those odds are reversed: Jack suited will win Why does this happen? The probabilistic outcomes change again if we pit these three hands against each other, two at a time.
In that case, sixes are slightly more likely to beat Ace-7, and Ace-7 is likely to beat Jack suited, but Jack suited is likely to beat a pair of sixes. The sixes are the best starting hand all by themselves. For one of the latter two to win, favorable community cards must appear on the flop, turn, or river.
The only way for the Ace-7 to beat paired sixes is for either an ace or a seven to turn up — or, less likely, for just the right combination of four cards to land on the board to make a straight or flush. Pit those same sixes against Jack suited, and the situation is reversed. In that scenario, there are more ways for Jack suited to improve. So Jack suited will usually beat a pair of sixes. For instance, if four more suited cards come up, the Jack suited will have a flush, but the Ace-7 will have a higher flush, and will win the hand.
Poker is a very complicated game, even more so once you add in player behavior during the various rounds of betting. If determining the edge and the odds were all it took to succeed at poker, probability theory would suffice, and one could fairly deem it gambling.
If it were a purely logical game like chess, it would merely require impressive feats of calculation to determine the winning series of moves. There may not be a single answer. As Harvey put it in the Discover article: Poker is like quantum mechanics. In chess, there is only one right move. In poker, there is a probability distribution of right moves. Like Poker for Chocolate: At times like that, I need to pay less attention to the math.
I found this enlightening analysis over at Cardplayer. Note the very specific circumstances described throughout: He just insists on seeing the flop. The optimal strategy is probably to make a small raise, both building a pot and disguising your hand. The optimal strategy would still win you money but against bad players, other strategies might win you more money. But when we can find ways to do better than optimal strategy against certain players, we do it. Theory of Games and Economic Behavior.
It offered an intriguing insight into the art of the bluff: Indeed, there are rare cases where game theory dictates you should fold pocket aces before the flop when playing a tournament.
In non-tournament play, the goal is not just to win the hand but to make the most money. In a tournament, you want to outlast your opponents to win it all. That might entail intentionally opting not to maximize your monetary gains on one specific hand to remain competitive in the tournament. You sacrifice short-term gain to achieve the long-term goal.
I once played poker with a group that included Harvey. So when I scored with pocket aces and nothing but rags low cards of varying suits after the flop, I pushed all-in, going heads-up with Harvey.
He correctly analyzed his chances, based on my all-too-predictable style of play. The optimal strategy can also depend on what type of poker is being played: The online software can analyze thousands of hands being played at the same time, and that larger sample space makes for a more accurate statistical analysis.
You can follow his exploits on Twitter: The mathematicians have had a good run when it comes to analyzing poker, but the Time Lord is rather cheekily on record predicting that physicists will prove to be the better poker players in the future.
He also had a corollary: Poker never lacks for suspense: Harvey once faced just that scenario — and a third queen appeared as the very last card.