Calculating Odds

Nick

Nick

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Administrator
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...
Outs The number of cards left in the deck that will improve your hand.
"I had four hearts on the turn, so I had only 9 outs left to finish that flush." Pot Odds The odds you get when analyzing the current size of the pot vs. your next call.
"There's $200 already in the pot, and only another $10 bet coming at me, so my pot odds are good if I hit that flush." Bet Odds The odds you get as a result of evaluating the number of callers to a raise. "With a 1 in 5 chance of hitting it, and knowing all six of these guys are gonna call my bet, my bet odds are good too." Implied Odds The odds you are getting after the assumed result of betting for the remainder of the hand. "Since I think these guys are going to call on the turn and river, my implied odds are excellent."

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).

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!

Article written by texasholdem-poker.com
 
J

johnph77

Enthusiast
A good source of further information on this subject can be found at wizadofodds.com
 
Chatmaster

Chatmaster

Guest
Nick congratulations, this is one of the best explanations I have read in a long time! Nicely done, even a complete idiot like me could understand that :D
 
M

MaxMavric

Guest
Is there a cliff notes version?

lol jusk kidding yea this helped a lot thx!
 
M

Mike Church

Guest
That's a good article. Yes, pot odds are important; no poker player can ensure a good night every time. A good poker player on $5/$10 might have a mean expectation of $10 per hour and a standard dev. of $100. In a 6-hour night, he only has a 59% chance of coming out positive, but over many nights he should rise. You can only win at poker if you take positive-expectation moves and reject negative-exp. opportunities (even when tempting).

Fallacies I've seen a lot of beginning poker players make include:

  • Overestimating long-shot probabilities (a common human fallacy, since the human brain can't really get itself about how unlikely, say, a 600:1 shot is),
  • failing to observe other players and model their strategies,
  • bluffing too often and failing to understand the true purpose of bluffing (which is to make other players call when you do have a hand; it's not about the free blinds),
  • calling when they shouldn't, to "keep people honest" or for curiosity's sake,
  • defending the SB/Bring-in on a junk hand like 84 off-suit.
Where computations get tricky, as the article points out, are the bet and implied odds. Modelling subjective probabilities (i.e. probabilities related to human strategy that can only be observed, not computed) is very tricky, I find.
 
ChuckTs

ChuckTs

Legend
too.... many ..... numbers

my friggin brain hurts
but good site nontheless
 
J

johnph77

Enthusiast
By calculation there are 133,784,560 possible 7-card poker hands.

The odds are 1::17 that you will catch any pair as your hole cards and, by inference, the chances those cards will be aces is 1::221.

On the other hand, mathematics, calculations and odds will least affect play in Texas Hold-Em than almost any other poker game. HE is a psychological game more than anything else. With only two cards hidden in each hand calculation of the odds of catching a particular card is relatively easy whereas in, say, 7-Card Stud, there are more cards visible and the calculations can become more difficult, what with having to remember who folded, how many cards they had exposed, what those cards were, etc. I'm not saying that psychology doesn't enter in Stud play, but the effect isn't as great as it is in HE.

gl

john
 
Nick

Nick

CC
Administrator
By calculation there are 133,784,560 possible 7-card poker hands.
That's quite a calculation you have there, I hope you didn't spend valuable time working it out... if you did I would be curious as to how long it took you to come up with such a large number.

What you mentioned about odds not being so important in Holdem could be argued against, although mathematically there's more chance of each card coming up... there is only 3 chances for a card to appear whereas in 7 card poker (not that I've played it) there is 5 chances is there not?
 
Ione

Ione

Enthusiast
bluestater said:
i think you gotta go with your gut in teh end
Gawd, I hope you're right. I'm a child of the Fifties, where girls and math just didn't mix... my brain hurts trying to wrap it around these concepts. But I refuse to give up, because everything I read says one should know something about these odds. Nevertheless, I tend to agree with you. There's something to be said for intuition. :cool:

BTW, thanks, Nick, for posting this. It helps to have it spelled out this way!
 
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X

xdmanx007

Legend
johnph77 said:
By calculation there are 133,784,560 possible 7-card poker hands.

The odds are 1::17 that you will catch any pair as your hole cards and, by inference, the chances those cards will be aces is 1::221.

On the other hand, mathematics, calculations and odds will least affect play in Texas Hold-Em than almost any other poker game. HE is a psychological game more than anything else. With only two cards hidden in each hand calculation of the odds of catching a particular card is relatively easy whereas in, say, 7-Card Stud, there are more cards visible and the calculations can become more difficult, what with having to remember who folded, how many cards they had exposed, what those cards were, etc. I'm not saying that psychology doesn't enter in Stud play, but the effect isn't as great as it is in HE.

gl

john
Limit poker regardless of version is 90 percent about numbers bluffing and what not are only mildly profitable skills in lmit. NL is the only style where you could argue that not needing to precisely know the numbers is acceptable, but you better have a general idea when you play NL or you WILL go broke!
 
S

superkmac72

Guest
Nick i was wondering if you knew what the odds of hitting a flush draw or any straight draw when you hold 67 suited. I love small suited connectors and i was wondering if any1 could give that answer.

Nick i was wondering if you knew what the odds of hitting a flush draw or any straight draw when you hold 67 suited. I love small suited connectors and i was wondering if any1 could give that answer.

sry bout the screwed up post, dont know what happened
 
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RammerJammer

RammerJammer

Visionary
Awards
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The simple* odds for hitting a flush on the draw when holding suited connectors:
Flop: 119-to-1
Turn/River: 1.86-to-1

The simple odds for hitting a straight on the draw when holding suited connectors:
Flop: 75-to-1
Turn/River: 0.85-to-1

The simple odds for hitting a straight flush on the draw when holding suited connectors:
Flop: 19,599-to-1
Turn/River: 10.9-to-1

- from Mike Caro's tables in "Super System"

*Probability of cards emerging from a standard 52-card deck. Does not take into account number of players, card rank, number of cards previously dealt in hand, etc.
 
Prysm

Prysm

Guest
Any easy way to calculate your % of making a hand is multiplying your outs by 4 for the turn and 2 for the river. Nice, quick, and accurate. Only problem I have with pot-odds and calculating my outs is keeping my outs live >_<
 
M

marcocroc

Guest
in the tournaments the best odds is about chipps in the table.If you have good hand and there are many chipps off others players you shoul call.Thats what i do and when i win i win a lot.Dont risky mutch money in lower chipps.
 
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