Holdem Odds | Advanced Poker Strategy : An Explanation of Hold 'Em Odds |
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Probability is a huge factor in texas hold 'em. Players use odds to determine their actions. The chances of finishing a flush or a straight, the probablity of getting an overcard, the percentage of times you're going to flop a set to match your pocket pair are all important factors in poker. Knowledge of these statistics is key to winning. In online games especially with very few (if any) tells, statistical knowledge becomes the main factor when choosing whether to bet, call, or fold. Here are some terms that you'll hear on this site and whenever you're talking about poker odds...
In Texas Hold 'Em, you commonly use outs and pot odds the most. This is also the starting point for those who want to learn about poker odds. To those out there who "ain't good at countin' much", you better get good because that is how it's done. At this point it's only simple division The numerator will be the number of outs you have. The denominator is the number of cards left that we haven't seen. The result will be the percentage chance of making one of those outs. Therefore, the most math you'll be doing will be dividing small numbers by 50 (pre-flop), 47 (after the flop), or 46 (after the turn). Click here for a series of examples on this. Before we move on, I must clarify one thing. A lot of you might wonder why we never factor the opponents' cards or the burn cards when figuring out how many cards are left. The reason is that we only consider "unseen cards". If you saw what the burn cards were, or an opponent showed you his hand, you would know that those cards are not going to be drawn and could use that. We typically do not know what they have, so we don't even think about it when talking about odds. For instance, take a standard deck of 52 cards, remove 2 Aces and burn 25 of them. If you drew the next card, what are the chances of it being an Ace? It would be 2/50 (2 Aces left out of 50 unseen cards). It would NOT be 2/25 just because you burned half the deck. Okay, do the same thing again, but this time you get to look at the burn cards. Let's say that of all the cards you burned, none were an ace. Now your odds are 2/25 because there are still 2 Aces and now only 25 "unseen cards". By that same reasoning, let's play a game of draw poker where you get 5 cards as usual, but your opponent gets 40. Say you got Ace, King, Queen, Jack all of Spades!, and a Four of Clubs. You get to ditch the Four and draw one from the remaining pile of 7 cards. What are your chances of getting that Ten of Spades? Assuming you don't get to see your opponents hand, your chances of drawing that card would be 1 in 47 (1 Ten of Spades in the deck, 47 "unseen cards"). It would NOT be 1 in 7. Let's say your opponent goes to the bathroom, and you cheat and look at his hand while he's on the crapper. If he doesn't have that Ten of Spades, that would be 1 in 7. If he did, well...it'd be 0 in 7. Pot odds are as easy as computing outs. You compare your outs or your chance of winning to the size of the pot. If your chance of winning is significantly better than the ratio of the pot size to a bet, then you have good pot odds. If it's lower, then you have bad pot odds. For example, say you are in a $5/$10 holdem game with Jack-Ten facing one opponent on the turn. You have an outside straight draw with a board of 2-5-9-Q, and only the river card left to make it. Any 8 or any King will finish this straight for you, so you have 8 outs (four 8's and 4 K's left in the deck) and 46 unseen cards left. 8/46 is almost the same as a 1 in 6 chance of making it. Your sole opponent bets $10. You if you take a $10 bet you could win $200. $200/$10 is 20, so you stand to make 20x more if you call. 1/6 higher than 1/20, so pot odds say that calling wouldn't be a bad idea. Another clarification...a lot of players want to somehow factor in money they wagered on previous rounds. With the last example, you probably had already invested a significant portion of that $200 pot. Let's say $50. Does that mean you should play or fold because of that money you already have in there? $50/$200? That's a big no. That's not your money anymore! It's in a pool of money to be given to the winner. You have no "stake" in that pot. The only stake you might have is totally mental and has no bearing on hard statistics. The next step is to use bet odds and implied odds. That's tougher, because it involves predicting reactions of other players. With bet odds, you try to factor in how many people are going to call a raise. With implied odds, you're thinking about reactions for the rest of the game. One last example on implied odds... Say it's another $5/$10 holdem game and you have a four flush on the flop. Your neighbor bets, and everyone else folds. The pot is $50 at this point. First you figure out your chance of hitting your flush on the turn, and it comes out to about 19.1% (about 1 in 5). You have to call this $5 bet vs a $50 pot, so that's a 10x payout. 1/5 is higher than 1/10, so bet odds are okay, but you must consider that this guy's going to bet into you on the turn and river also. That's the $5 plus two more $10 bets. So now your facing $25 more till the end of the hand. So you have to consider your chances of hitting that flush on the turn or river, which makes it about 35% (better than 1 in 3 now), but you have to invest $25 for a finishing pot of $100. $100/$25 is 1 in 4. That's pretty close. But there's more!... if you don't make it on the turn, it'll change your outs and odds! You'll have a 19.6% chance of hitting the flush (little worse than 1 in 5), but a $20 investment for a finishing pot of $100! $100/$20 is 1 in 5. So the chances would take a nasty turn if you didn't hit it! What's makes it more complicated is that if you did hit it on the turn, you could raise him back, and get an extra $20 or maybe even $40 in the pot. I'll let it go at that, as once you've mastered simple outs and pot odds, bet and implied odds are just a longer extension of these equations. If you sit and think about these things while you play, it'll come to you eventually without any tutoring. Good luck! The Easy Example: A pocket pair You start with a pair of Jacks in the pocket. Not too shabby. The flop however, doesn't contain another Jack. Lesson 1 : What's my chance of getting a Jack on the turn? You need to just figure out the number of outs and divide it by the number of cards in the deck. There's 2 more Jacks. There's 47 more cards since you've seen five already. The answer is 2/47, or .0426, close to 4.3%. Lesson 2 : No luck on the turn, how 'bout the river? Still 2 Jacks left, but one less card in the deck bringing the grand total to 46. What's 2/46? That's .0434, which is also close to 4.3% Your chances didn't change much. Lesson 3 : Screw getting just one Jack! I want them both! What are my chances?! Since we're trying to figure out the chances of getting one on the turn AND the river, and not getting one on EITHER the turn or river, we don't have to reverse our thinking. Just multiply the probability of each event happening. Chances of getting that first Jack on the turn was .0426, remember? The chance of getting a second Jack on the river would be 1/46, because there'll only be one Jack left in the deck. That's about .0217, or 2.2%. To get the answer, multiply 'em. .0426 X .0217 is about .0009! That's around one-tenth of a percent. I wouldn't bank on that one. Lesson 4 : Hey, what were my chances of getting a pair of Jacks anyway? To figure that out, think of it as getting dealt one card, then another. What are your chances of the second card matching the first one? There will be 3 cards left like the one you have. There's 51 cards left in the deck. 3/51 is .059 or 5.9%. What the chance that it'll be Jacks? Well, there's 13 different cards. So, .059/13 is about .0045, a little less than half a percent. Lesson 5 : What were my chances of getting a Jack on the flop? Now you do have to "think in reverse" as in the previous example. Figure out the chances of NOT getting a Jack on each successive card flip. First card you have a 48/50 chance (48 non-Jack cards left, 50 cards left in the deck), second card is 47/49, third card is 46/48. Those come out to .96, .959, and .958. Multiply them and get .882, or an 88.2% chance of NOT getting any Jacks on the flop. Invert it to figure out what your chances really are and you get .118 or 11.8%. This will be your chance to get one or two Jacks. Example #2 "The straight draw" You start with a Jack of Spades and a Ten of Spades. You get a rainbow flop with a Queen of Spades, a Three of Diamonds, and a Nine of Clubs. You've got a straight draw. Lesson 1 : What are my chances of hitting it on the next card? Same as before, but with different outs. A King or an Eight will complete your hand. There are presumably four of each left in the deck. You've got 8 outs. The chance of getting one of them on the turn is 8 over 47, because there's 47 cards left in the deck. That comes out to about .170, or around 17%. Lesson 2 : I didn't get it on the turn! What are my chances now!? There's still 8 cards left in the deck that'll help you, but 46 cards left in the deck. That's 8 over 46. It changes to .174. It's improved to a whopping 17.4%! Lesson 3 : I should of thought about my total chances first, I'm such an idiot. What are my chances of getting that card on the turn OR the river? Once again we'll have to calculate the chances of a King or Eight NOT appearing, so we can do it like the last problem (in this case, {39/47} X {38/46}). Or, since we've already figured out our chances in the previous two lessons, we can just invert the probabilities and multiply 'em. You had a .170 chance on the turn, and a .174 on the river. By inverting, I mean subtracting them from one. Now we've got .830 and .826! Multiply and get .686! That's our chance of NOT hitting our card at all. So invert it again and get .314, or 31.4%. Example #3 "Top two pair" You get dealt a King of Diamonds and a Nine of Hearts. The flop is lookin' pretty good for you with a King of Spades, a Nine of Clubs, and a Four of Clubs. Top two pair! Lesson 1 : What are my chances of getting a full house on the turn? To get a full house, you need another King or Nine to pop up. There are presumably two of each left in the deck. So you've got 4 outs. After the flop there's always 47 cards unaccounted for. 4/47 is around .085 or an 8.5% chance of you getting that boat. Lesson 2: What are my chances of getting a full house on the river? If it didn't happe n on the turn, your chances usually don't change all too much, but let's check. You've still got 4 outs and now 46 unseen cards left. 4/46 is about .087 or around an 8.7% chance of hitting it on the river. A .2% difference. Sorry. Lesson 3 : How about the chances of getting the boat on the turn OR the river? Like the previous examples, to figure your chance of something happening on multiple events, you need to calculate the chance of it NOT happening first. On the turn it won't happen 43/47 times. On the river it won't happen 42/46 times. 43/47 is .915, and 42/46 is .913. Multiply them and get .835, or 83.5% chance of it not happening. Invert that and you get a 16.5% of getting at least a full house by the showdown. Lesson 4 : What do you mean by "at least"? Since we figured the chances to NOT get dealt a full house, the chances are built in if the turn and river are two Kings, two Nines, or a King and a Nine. If you are dealt two cards both of either King or Nine, it'll be four-of-a-kind and not a King and Nine 33% of the time. Think of it as being dealt one card then the other. What are the chances of the first card matching the second? Whether it's a King or Nine, there will be only one unaccounted for, but two of the other. That's 1/3, or 33%. Lesson 5 : Then what are my chances of getting four-of-a-kind? This is a little more abstract. I hope I warmed you up for this with the previous lesson. |
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