Okay, now that everybody has read the Hand Odds article and can quickly come up with usable hand odds it’s time to explain exactly why you would want to do this. It’s obvious that knowing the probability of making a hand is useful for deciding hand strength (a topic that I plan on discussing in more detail at a later date); the great thing about it is that you can actually use those hand odds in conjunction with pot odds to arrange it so that – at least over extended time periods – you can guarantee a positive return for your investments. And remember that … you need to look at any money that you put in the pot as an “investment”. Nobody ever wants to invest money in an “opportunity” that will not pay off so why would you do that whilst playing poker? You most definitely shouldn’t (there’s always the exception of the bluff and semi-bluff, but those are some more advanced topics that I will be covering later) and, if you correctly utilize the pot odds vs. hand odds technique that I describe below, you won’t.To better explain this concept, let us discuss a hypothetical. Let’s say that you run into somebody that wants to do a bit of gambling but wants to keep it extremely simple – he offers to pay you $5 if you risk $1 on a roll of one die. All you need to do is pick the number that the die (a typical 6-sided die) will land on – if you’re correct, he pays you $5; if you’re wrong, he keeps your $1 bet. Do you take the bet?
Since the payout (the pot odds) is only 5 to 1 on your bet and the probability of you picking the right number and winning (the hand odds) is only a 1 in 6, it should be rather obvious that you should choose not to play with this guy. Consider if you chose to take him up on his bet and rolled the die 120 times. You have a 1 in 6 chance of picking the correct number on each roll of the die which means that you can expect to win the bet 20 times (1/6 of 120). Of course what that means is that you can expect to “win” $100 (20x$5) but would have paid $120 (120x$1) for this prize – a net loss of $20.
By using hand odds vs. pot odds it’s easy to see that, before rolling dice with this guy, he needs to agree to pay at least $6 if you win. That would be “even odds” (pot odds would be 6 to 1 and the hand odds would be 1/6). You actually shouldn’t take this bet either as the return on your investment in this situation is going to average out to a net gain of zero. You’ll pay $120 for 120 rolls and should win the bet 20 times resulting in a positive winning of $120 – all you’ve managed to do is come out even… If you decide to play with the even odds situation, the only thing that is going to decide whether or not you come out ahead is the noise fluctuation on the probability – often referred to as “luck”. You see, that’s all luck is – the mathematical noise seen in probabilities run on limited sample sizes. If you ever decide to take a bet with even odds, you’re basically betting on the mathematical noise and this is always unknowable. You would be counting on luck and luck is something that you should never count on…
To complete the above example, let’s say this guy agrees to pay $7 on a correct guess and a roll still only costs $1. Since the pot odds have now become 7 to 1 and the hand odds haven’t changed (1/6), this is the game you want. 120 rolls will still cost you $120; however the 20 times that you will win will bring in $140 – a net gain of $20. The object here is to make sure that the payout justifies the bet.
Now finding somebody willing to pay you more than $6 for the correct guess of a die roll is not an easy thing to do. This whole concept of pot odds vs. bet odds is actually very easy to understand in a one die roll situation and finding a person that doesn’t understand this would be difficult. In poker, however, many other variables exist (unknowns, machismo, greed…) and it is often amazing how often you can find people willing to take these unfair bets. Pay attention and use this to your advantage.
To illustrate my point, let’s now look at a poker situation. Let’s say you’re holding a ten of hearts and a jack of hearts at a $2/$5 limit Texas Holdem table. There are five people sitting at the table and you are on the button. The first two have limped in and you decide just to call the $2. The small blind calls and the big blind checks. The flop is shown and it is an ace of hearts, a nine of spades and a deuce of hearts. Positions 1 and 2 check, position 3 bets $5 and position 4 calls. Do you call the $5 and stay in the hand?
Let’s analyze the situation. You’re currently holding nothing – an ace high with kickers of jack, ten, nine and two. Of course everybody has at least an ace high making your hand a loser if anybody holds merely a king or queen; however, you are holding a four-card flush. Now, since you are able to calculate your hand odds at making the flush to be about 1/3 on the next two cards (I explained this same situation in my Hand Odds post) it’s time to calculate the pot odds and see if this call is justifiable – there is currently $20 in the pot (and two players – the initial checks – left to act) making the pot odds at least 4 to 1. Since the pot odds are slightly better than the hand odds, you decide to call the $5 bet.
Position 1 folds and position 2 raises to $10 (a check-raise – usually a sign of some sort of “made” hand). Position 3 calls and position 4 folds. The action is now on you and you need to call the $5 raise in order to stay in the hand. What do you do here?
Okay, since the hand hasn’t changed your hand odds are the same (1/3). The pot, on the other hand, has changed – there is now $40 in the pot. The pot odds have rather interestingly increased to 8 to 1 for your $5 call. Since your hand odds are still 1/3 you see that this might be a good place to raise. Because you’re sitting at a limit game, the most you can raise is $5 which would then pull the pot odds down to 4 to 1 on your $10 bet. A 4 to 1 payout is justifiable for your hand (remember, your hand odds are still 1/3) so the correct play here is actually to raise. You raise. (note: You just made a mistake - please read my added comment at the end of this post!!!)
Position 2, the original check-bet then re-raises. Position 3 folds and the action is once again on you. Okay, so position 2 is representing a decent hand. I would guess that he’s holding at least an ace. Of course, for us this is good information – a flush beats whatever his “made” hand is. There’s always the possibility that he’s playing the same draw that you are but that’s more of a topic for hand strength that I will be covering later. Currently you are being asked to act on position 2’s raise. Now, being that this is a limit game and pre-river raises are usually limited to three per round, your options are only to call or fold. Your hand is the same (a 1/3 probability flush draw) and the pot has grown to $60 making your pot odds an amazing 12 to 1. You call the $5.
The turn then gets dealt and it’s the five of clubs. Position 2 bets the maximum of $5. Well, the five of clubs was no help to you – you still don’t have a made hand and your hand odds have now dropped to 1/5 for making the flush on the river. How do you play this bet? The pot is now $70 making your pot odds 14 to 1 on a $5 call. It also makes your pot odds 7 to 1 if you decided to raise which is still justifiable. You raise the bet and put $10 in the pot.
Position 2 quickly re-raises your raise and the action returns to you. There’s now $90 in the pot – an 18 to 1 pot odds for a $5 call or a 9 to 1 pot odds if you were to raise. You raise with your 1/5 hand odds. Position 2 calls.
The river is dealt and it’s the king of hearts. The pot has $105 in it and you’re holding the second nut (a queen of hearts with any other heart beats you, but that’s it). Position 2 bets $5 and the action is now on you. You made your hand and position 2 has been representing a made hand from the flop – of course any hand that was made on the flop is a loser to your hand now. You raise the bet to $10. Position 2 calls (he sees the three hearts on the board and realizes that he might be beat) and shows his hole cards – a pair of aces. You display your hole cards and rake in the $125.
Now an interesting aspect of the above example hand is seen by taking a more high-level view of exactly what happened there. Bottom line on that hand is that you won a $125 pot by risking $50 of your money. You also did this by playing a hand with 1/3 odds of improving after the flop. You got a 2.5 to 1 payout with a 1/3 probability which, statistically speaking, is a bad play. The thing to remember is that you got this win by correctly playing the hand odds vs. pot odds on all of your action. (note: That last statement is WRONG! Please read my added comment at the end of this post!!!) There are several other aspects that would have changed your play and I will be covering these in later articles. For now, however, I just want you to understand hand odds vs. pot odds and I believe that this example hand is a good study.
There’s really nothing better than experience for learning how to win at poker; however, if the time is taken to understand your hand odds vs. the pot odds, your experiences will have much more meaning to you. It’s one of the basic tools that a poker player needs to understand if he wants to become a winning poker player.
bis später,
Coriolis
Okay, I think I’ve found my problem with the above example. I mentioned that it was interesting that you were able to get a 2.5 to 1 payout on a 1/3 hand odds and I believe I have figured out what I did wrong here. The problem was that I was “giving up ownership” of my chips too quickly. Or, to put it another way, I was calculating my pot odds incorrectly – rather than using the total amount of money for the betting round, I was only calculating my odds based on the additional bets made to the pot. I’ll go through that hand again and explain…
ReplyDeleteOkay, so five players were in for the flop which put $10 in the pot. The flop hit (Ah9s2h) and we were holding Jh10h. Positions 1 and 2 check, position 3 bets $5 and position 4 calls. Do you call the $5 and stay in the hand?
Let’s analyze the situation. You’re currently holding nothing – an ace high with kickers of jack, ten, nine and two. Of course everybody has at least an ace high making your hand a loser if anybody holds merely a king or queen; however, you are holding a four-card flush. Now, since you are able to calculate your hand odds at making the flush to be about 1/3 on the next two cards (I explained this same situation in my Hand Odds post) it’s time to calculate the pot odds and see if this call is justifiable – there is currently $20 in the pot (and two players – the initial checks – left to act) making the pot odds at least 4 to 1. Since the pot odds are slightly better than the hand odds, you decide to call the $5 bet.
Position 1 folds and position 2 raises to $10 (a check-raise – usually a sign of some sort of “made” hand). Position 3 calls and position 4 folds. The action is now on you and you need to call the $5 raise in order to stay in the hand. What do you do here?
So far everything is the same as the example above – here is where things change…
Okay, since the hand hasn’t changed your hand odds are the same (1/3). The pot, on the other hand, has changed – there is now $40 in the pot. The pot odds have rather interestingly increased to 8 to 1 for your $5 call. The problem here is that a $5 call is actually not a $5 call. You have already put $5 in the pot on this bet so, if you call the $5 raise, you’re actually going to be putting $10 in what is now a $40 pot – not to mention that, since your first $5 is already in that pot, you’re only looking at a return of $35 for your $10 bet. What this comes out to, of course, is a 3.5 to 1 pot odds situation on a 1/3 hand odds. It’s still good, so rather than the miscalculated raise that I mentioned above, the correct thing to do here is merely call. You call the raise and throw another $5 in the pot.
The good thing for us here is that position 2 never gets the chance to re-raise since we actually stopped the betting round with our call. The turn is then dealt (5c). Since I had position 2 holding the bullets in this example, he is very likely going to bet the maximum $5 on his action and we’ll say that this is where position 3 decides to leave the action and folds. The action comes to us and we need to make a decision on this $5 bet. Since we didn’t hit our flush on the turn, our hand odds have dropped to 1/5; however the $5 bet from position 2 has increased the pot to $50. What do we do here?
Simply put, the pot odds are now 10 to 1 on our $5 call and 5 to 1 on a raise to $10. We raise the bet to $10 playing the pot odds. Of course the set of aces re-raises our raise which then brings the pot to $65. Do we call this raise?
This is where we would need to bow-out. The pot odds are 13 to 1 on a $5 call – unfortunately for us, we’re already in for $10 on this round of betting. What this means, of course, is that we are now being asked to shell out $15 on a $65 pot (and we can subtract this round’s already placed bet of $10 from the potential profits bringing the pot odds down to 11 to 3 – $55 payout on a $15 bet). This, of course, is not good and we would then fold and give the pot to the set of aces. What happened here is that position 2 played his bets correctly and took the $65 pot ($43 profit) by getting us to fold. He got very close to a 2 to 1 return on his $22 hand investment. We paid $22 of that return but opted out when the odds turned against us in an effort to cut our losses.
Let’s have position 2 get a bit greedy here and attempt a slow-play on our raise and see what happens…
On the turn, if position 2 merely called our raise, the pot would be at $60 and the river would be dealt (Kh – board is Ah9s2h5cKh – we hit our flush). Position 2 was being greedy by not raising and it cost him the pot. We’re in for $22 and very likely to win a minimum of $60 ($38 profit) although we are now free to attempt to increase this profit by trying to get position 2 to put more money in that pot. Or minimum return on our investment here is 38 to 22 or approximately a 1.7 to 1 return but since we hit our hand, we’re free to work on that.
Let’s say position 2 still thinks his set is best after seeing that heart hit and bets another $5. The pot is now $65 and, since we hit the hand, we raise. Position 2’s not an idiot, so he calls (rather than re-raising) our raise and loses the showdown. We rake in $80 from the pot ($48 of it profit). The return on investment here is 48 to 32 or 1.5 to 1 – lower return, but more profit. We were able to get this by making our hand.
Interesting… This hand is actually a very good example of “cutting your losses”. The thing to remember here is that, once the turn was dealt, you are going to lose this hand 80% of the time (4/5). However, by playing the odds correctly, you are limiting your losses to $22 (as opposed to the $42 that you would have lost if the flush didn’t hit as I originally described this hand…).
Sure, you’re limiting your profits a bit (getting at most a 1.7x return as opposed to the 2.5x return shown in the original example); however, since you are only going to take this hand 20% of the time I think the limit to your losses is far more important. After all, 80% of the time you will end up paying. (Of course 20% of the time you will get your flush on the turn, so you’re actually only going to be losing this hand 80% of 80% of the time or 64% of the time – about 2/3…)
So, what I’ve shown here actually breaks down as follows… About 64% of the time, we are going to lose $22. On the other 36%, we are going to bring in a minimum profit of $38 (more likely $48 as described with the bad final bet by position 2 above) – assuming the hand plays out to the river. You should notice that that comes out to bringing in an average profit of $43 on our $22 bet approximately 1/3 of the time and losing the $22 bet 2/3 of the time. Finally this is making sense as it now pretty much demonstrates an even odds situation (there’s a dollar missing on the profit side but I believe we’d be bringing in the $48 much more often than the $38 which would easily make up for this).
Using my original (and incorrect) play, you would be losing $42 64% of the time and bringing in an minimum profit of $65 (pot size [$105] minus our investment [$50]) which is more likely to be $75 if position 2 bets after the river. This is bringing in an average profit of $70 on our $42 bet approximately 1/3 of the time and losing the $42 bet 2/3 of the time. Since our profit is less than double our risk, the odds are no good.
Bottom line here is the mistake I made in my original example was to not include the total amount of a bet being made each round when calculating pot odds. I erroneously calculated the pot odds based only on any forced additional betting. I believe that the above analysis demonstrates that full betting amounts need to be considered for action on each betting round.
Sorry about the confusion here, but I think this has ended up becoming a very interesting analysis. All that needs to be taken from this is that you should never calculate pot odds using incomplete data.
bis später,
Coriolis