The Curse of Pot Odds

pokeroff 0 1159
The Curse of Pot Odds

Using Pot Odds is a way of estimating the value of your draw. It's a way of balancing the odds of you completing your hand with the amount of money you stand to win. There is a whole lot of difference in staying in a hand by paying $5 to win $10, and playing $5 to win $100 – pot odds is a way of working out whether you should stay in a hand or not.

Let's work through a simple example.

Let's say we had  Q♦J♦ as our pocket cards, and we made a 3-bet on an opponent, and he called. The flopped turned out nicely for us, as K♦T♦2♣ so we bet $2.50 to bring the pot up to $4.35, and our opponent called. The turn came out 5♠

For our opponent to have stayed in the hand, his range of cards must be tight. We can deduce to stay in the hand he must have one of the following sets of pocket cards:

  •  JJ - 3 combinations
  • KQ - 2/3 of all the combinations
  • AK - 1/2 of all the combinations

You can make allowances for this range, but they will not influence our evaluation of the hand, and now we will explain why.


Case #1

Assuming that on the turn we only have $7 left in stakes, and there is $9.35 already in the pot. What choices do we have? Raise, or bet. Most players will choose raise, with a straight-flush draw. Let's see if this is correct.


We bet $7 into the $9.35 pot. JJ should always fold, and all the other hands should call, therefore our fold equity is 20 percent, whereas our equity against the call range is 32.74 percent. This leads us to the following calculation:


EV = 20% * 9.35$ + 80% * [(9.35$ + 7$ + 7$)*32.74% - 7$] 

EV = +2.315$ 


The good news is our bet is likely to be profitable, even though we bet with the worse hand on the turn, and we have yet to address the question concerning the profitability of our action.


Case #2


The above is generally the answer as to what to do when faced with short stakes, and you have a good draw. Making a bet with a good draw is almost always a suitable play. But now let's imagine we have not $7 as our stake, but $15. What should be do now?


Typically 99 percent of players asked will say 'BET!!!” - but they would be wrong!


We really cannot make a bet, even with our excellent straight-flush draw, and this is because of the pot odds. Let's make this clear:


The situation is as follows: We make a $6 bet into the $9.35 pot. The probability is that the opponent drops JJ is exactly 20 percent. However, let's presume he's clinging on to the hope of completing his set and 10 percent of the time will call. That leaves 70 percent of cases in which we are raised.


The first two points have to be totally clear. Either we take the $9.35 pot without a fightback, or we will have a certain expectation in the form presented. But what do we do when we are raised?


We make a $6 bet, but our opponent raised up to $15. We now have to give up $9 to call, whilst the pot if we call will be $39.35 – this gives us pot odds of 22.87 percent (9 / 39.35) to make the call.

Upon this scenario most players will say “we have almost 33 percent against his top pair! If our equity is larger than the pot odds, we should make a call, as it is profitable!”



Let's take a few steps back. What's your probability if you don't make the call and fold instead? It is equal to zero? Nope – you've already contributed $6 to the pot, which you will lose. Expectation of fold is only equal to zero when we take into account EV from a single action (call all-in) and not from the whole line on the turn (bet and call all-in). Only in the last case will there be a real impact to our bankroll.

In addition, the action you take now – raise or call – is profitable (pot odds show it clearly) only compared to folding, but the whole line turns out to be a loss. Here is a simple explanation:


The probability of calling is [($15 + $15 + $9.35) * 32.74% - $9] = +$3.88. That may be so, but it is only the expectation of that we will make if we call on the turn, when we will have already contributed $6 to the pot. We need to take this money into account when we calculate the total expectation of the bet-call line on the turn.


The full expectation of our actions on the turn in this case equals [($15 + $15 + $9.35) * 32.74% - 15%] = -$2.12.


As, you can see, that puts our supposedly monster draw on the negative end of the scale. It's not difficult to see that if we subtract our initial bet of $6 from the probability of call (+$3.88), we get the exact expectation of the whole line on the turn.


So, in reality, after our opponent's raise. We can choose between one of two actions:


  • Fold – we lose the $6 that we bet on the turn
  • Call – we will have part of our money returned, but we will still make a loss


So, the probability of making a call that we got on account of the pot odds on the turn (when we called $9) shows only how much to call is better than folding. If we fold, we lose $6, and if we call, we lose “just” £2.12. In other words, the pot odds tell us correctly that we need to call our opponent's all-in after betting on the turn, but to keep quiet about the profitability of the whole draw line.


This brings us to a important conclusion: Sometimes in poker we have to make a choice between two loss-making decisions and choose the one that minimises the loss.


Here is one further example. If the probability on a certain street really depended only upon the sum of the additional bet that we have to put into the pot after an opponent's raise (and, correspondingly, on the pot odds), we would have to bet 99.9 percent of our stake on every flop, and raise when we were on the blinds. That means our pot odds would be around two percent! This would also mean that every call would be extremely profitable, even with 7♠2♦. But obviously, there is something quite wrong with this!


As a result, we would end up with the following sad picture on the turn:


  • 20 percent of tries we bet and win $9.35 pot from a pair of Jacks
  • 10 percent we bet and get called (probability here is equal to [(6 + 6 + $9.35) * 32.74% - 6] = +$0.99
  • 70 percent we bet and get raised (probability of the right call equals -$2.12)

We multiply all the expectations and get the total probability from the bet-call line on the turn with our super monster draw:


EV = 20% * $9.35 + 10% * $0.99 + 70% * (-$2.12) 

EV = +$0.49


Thankfully, it ended up positive, though you will no doubt be surprised that if we altered the range just the tiniest bit and give ourselves a 15 percent fold equity and zero percent calls, the line already becomes unprofitable.


So, what happens if we check on the turn? Won't this option raise more profit than bet-call?


Indeed it will. If we check on the turn, it's the same as all five cards looking for one bet from the flop. Let's presume that our opponent's range on the flop is the same as it was before (AK, KQ, JJ). If they were any hands that he folded to on when bet, then this fold equity was the same as for the previous bet-call line, so we don't need to take that into account. On the flop against {AK, KQ, JJ} our equity is about 54 percent, so:


EV = ($4.35 + $2.50 + $2.50) * 54% - $2.50)

EV = +$2.55.


And so, without taking potential pay-outs into account, if we land an ace on the river we complete our straight (or straight flush if we're really fortunate) whilst our opponent may be generous in backing his own AK. It's not really necessary to explain such a preferable line.


So, what can be said in conclusion? First, as you have seen, even the most simple poker conceptions can become extremely complex if you calculate them exactly. Secondly, any budding poker player should be looking to learn such concepts and try to master them so the numbers can be successfully crunched inside one's skull. Thirdly, never take “standard” lines and understand them as the undeniable truth. There is always plenty to learn in poker, so always remain willing to learn – that way you'll always be open to making some interesting and valuable discoveries.