If you are serious about playing poker then there are two types of odds that you need to know about and how they relate. These are hand odds and pot odds. When working with odds and probabilities, you will be required to know at least some elementary algebra. We would suggest that if you have a grudge against math, then poker is not the game for you. You may simply choose to memorize the odds for each type of hand, but the person who is able to adapt and calculate odds on the fly will be a far better player.
Hand odds are the chance of making a particular hand. For example, if you hold 2 hearts in your hand and there are 2 hearts on the flop, your hand odds for making a flush by the river are 2 to 1 or approx 33%. This means that for approximately every 3 times you play this hand, you will hit a flush one of those times. If instead your hand odds were 3 to 1, then you would hit your hand 1 out of every 4 times. Your hands odds can be converted into a percentage by adding 1 and dividing, e.g. 2 to 1 becomes ( 2 + 1 ) / 1, which equals 33%.
X to 1 odds = You hit you hand 1 out of (X+1) times
X to 1 odds = 1 / (X+1) = % chance to hit your hand
How many outs?
In order to calculate your hand odds, you first need to know how many outs your hand has. Outs are defined as a card in the deck that helps to make your hand. For example, if you hold AK of hearts and there are 2 hearts on the flop, that leaves 9 remaining hearts that would complete your flush, since there are 13 cards of each suit. It may be the case that a number of the remaining hearts are held of have already been folded by other players, in which case there would be less than 9 remaining hearts in the deck. If you know for sure that someone is holding or has folded a heart then you can take that into account when calculating your odds, but in most situations you will not know, and therefore, must do the calculations only with the knowledge that is available to you; that knowledge being your hand and the cards on the table.
Don't overcount your outs:
When calculating your outs, it is important not to overcount your outs. An example of such a situation would be a flush draw in addition to an open straight draw.
You hold Jh Th and the board shows 8h Qh Kd. A Nine or an Ace gives you a straight, of which there there are 8 (8 outs), while any heart gives you the flush, of which there are 9 (9 outs). However, there is an Ace of hearts and a Nine of hearts, so you don't want to count these twice toward your straight draw and flush draw. The true number of outs is actually 15 instead of 17.
Outs which aren't true outs:
When drawing to a hand you always want to be drawing to the nuts, i.e. the best possible hand. The situation in which you will lose the most money is where you draw to and make a hand, but your opponent makes a better hand. An example would be an open ended straight draw when two of a particular suit (of which you have none) are on the table. Where you would normally have a total of 8 outs, 2 of those outs will result in three to a suit on a table, making a possible flush for an opponent. As such whilst you would have 8 outs to make your straight, you only really have 6 outs for a nut straight draw. These 6 outs are referred to as true outs. Below is an even more complicated example.
You hold J8o and the flop comes 9TJ rainbow (all of a different suit). To make a straight, you need a Queen or a 7, giving you a total of 8 outs. However, if a Queen drops the board is now 9TJQ. This means that anyone holding a King (a reasonable possibility in a big game) will have made a King high straight which dominates your Queen high straight. Therefore, the only card that really helps you is the 7 which gives you only 4 true outs.
Proper Calculation
Once you know how many outs you have on a hand, you can calculate your hand odds, i.e. what percent of the time you will hit you hand either on the turn, the river, or by the river (on the turn or river). When calculating the probability of a single event, e.g. hitting the out on the Turn or the River, this is simply: Outs / Remaining Cards. Calculating the probability of hitting the out on either the Turn or River is a little more complicated, and is done by finding the probability of you cards not hitting twice in a row. This is calculated as below:
Flop to River % = 1 - [ ((47 - Outs) / 47) * ((46 - Outs) / 46) ]
The number 47 and 46 represent the remaining cards left unknown after the flop and turn respectively. An example of a flush draw is shown below:
= 1 - [ ((47 - 9) / 47) * ((46 - 9) / 46) ]
= 1 - [ (38 / 47) * (37 / 46) ]
= 1 - [ 0.81 * 0.80 ]
= 1 - 0.65
= 0.35
= 35% chance of hitting your flush on either the turn or river
Most of the time we want to see the percentage in hand odds, the reason for which will become apparent once we have discussed pot odds. To change a percentage to odds, the formula is:
Odds = ( 1 / Percentage) - 1
Therefore, to change the 35% draw into an odd, we do the following:
= ( 1 / 35% draw ) - 1
= (1 / 0.35) - 1
= 2.86 - 1
= 1.86 or approx 1.9 to 1
Shortcut Calculation
We have described above how to properly calculate hand odds, but a shortcut does exist that will make it much easier to calculate odds on the fly. After you have found the number of outs you have, multiply by 4 and you will get a close estimate to the percentage of hitting that hand by the River. Multiply by 2 instead to get a percentage estimate for either the Turn or the River, i.e. a single event.
For example, for a flush draw with 9 outs, multiplying 9 by 4 you get 36%. This closely resembles the figure of 35% that we worked out above.
In order to get the ratio hand odds, rather than the percentage odds you use the same formula as above, although it can be rephrased slightly to make the math appear easier:
Odds = (100 / Whole Percentage) - 1
Using 100 divided by the whole percentage number, such as 35%, we can easily see that 100/35 is equal to about 3. We minus 1 from that and get a rough estimate of our odds at about 2 to 1.
Chart
Below is a chart showing the hand odds for making a hand on a single event (1 card) or with 2 cards (turn and river) depending on the number of outs.
| Outs | 1 Card (Turn or River) | 2 Cards ( Turn and River) |
|---|---|---|
| 2 | 22 | 12 |
| 3 | 14 | 7 |
| 4 | 10 | 5 |
| 5 | 8 | 4 |
| 6 | 6.7 | 3.2 |
| 7 | 5.6 | 2.6 |
| 8 | 4.7 | 2.2 |
| 9 | 4.1 | 1.9 |
| 10 | 3.6 | 1.6 |
| 11 | 3.2 | 1.4 |
| 12 | 2.8 | 1.2 |
| 13 | 2.5 | 1.1 |
| 14 | 2.3 | 0.95 |
| 15 | 2.1 | 0.85 |
| 16 | 1.9 | 0.75 |
| 17 | 1.7 | 0.66 |
Pot odds is simply the ratio of the amount of money in the pot compared with how much money it takes to call. For example, if there is $100 in the pot and it takes $10 to call, your pots odds are 100:10, or 10:1.
Pot odds ratios are a very useful tool to see how often you need to win the hand to break even. For example, if there is $100 in the pot and it takes $10 to call giving you pot odds of 10:1, you must win the hand 1 out of 11 times in order to break even. To play the hand 11 times will cost you $110, but when you win you get $110 ($100 + your $10 call).
The usefulness of hand odds and pot odds becomes apparent when you start comparing the two. As we calculated above, with a flush draw, your hand odds for making your flush by the river are approx 2 to 1. Let's say you are in a hand and its $5 to you on the flop to call. What should you do? Your first instinct should be: what are my pot odds?
If there is $15 in the pot plus a $5 bet from an opponent, then you are getting 20:5 or 4:1 pot odds. This means that in order to break even, you must win 1 out of every 5 times. We have already calculated that with a flush draw your odds of winning are approximately 1 out of every 3 times, so you should realized that not only will you break even, but should make a profit. Lets see what should theoretically happen if you were to play this hand 100 times from the flop, when it is checked to the river.
Net Cost to Play = 100 hands * $5 to call = -$500
Pot Value = $15 + $5 bet + $5 call
Odds to Win = 2 to 1 or 35%
Total Hands Won = 100 * Odds to Win (35%) = 35 wins
Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
= -$500 + (35 * $25)
= -$500 + $875
= $375 Profit
As you can see, you should be calling and playing this flush draw, because you'll be making money in the long run according to you hand odds and pot odds. Simply put, if your pot odds are greater than your hands odds then you should be making money in the long run by drawing to the hand.
The more often you calculate hand odds and pot odds your ability to memorize and calculate will improve, and it will you lead you to make many correct decisions in the future.
Many people forget that the theoretical calculations from the flop to the river assume that there is no betting on the turn. Therefore, whilst the hand odds for a flush draw are approx 2 to 1, you can only call a 2 to 1 pot on the flop if your opponent will let you see both the turn and river cards for one call. Most of the time this is not the case, and so rather than calculating odds from the flop to the river you should calculate them one card at a time. The odds of making you flush on any one card is approx 4 to 1. Look below to see an example of incorrect pot odds match:
You Hold: Flush Draw
Flop: $10 Pot + $10 Bet
You Call: $10 (2 to 1 odds)
Turn: $30 Pot + $10 Bet
You Call: $10 (4 to 1 odds)
Net Cost to Play = 100 hands * ($10 Flop call + $10 Turn Call) = -$2000
Pot Value = $10 + $10 Flop Bet + $10 Flop Call + $10 Turn Bet + $10 Turn Call
Odds to Win = 2 to 1 or 35%
Total Hands Won = 100 * Odds to Win (35%) = 35 wins
Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
= -$2000) + (35 * $50)
= -$2000) + $1750
= -$250 Profit
= -$2.50 / hand
Now look at an example of correct pot odds match:
You Hold: Flush Draw
Flop: $30 Pot + $10 Bet
You Call: $10 (4 to 1 odds)
Turn: $50 Pot + $16 Bet
You Call: $16 (about 4 to 1 odds)
Net Cost to Play = 100 hands * ($10 Flop call + $16 Turn Call) = -$2600
Pot Value = $30 + $10 Flop Bet + $10 Flop Call + $16 Turn Bet + $16 Turn Call
Odds to Win = 2 to 1 or 35%
Total Hands Won = 100 * Odds to Win (35%) = 35 wins
Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
= -$2600) + (35 * $82)
= -$2600) + $2870
= $287 Profit
= $2.87 / hand
As you can see, calling a flush draw with 2 to 1 pot odds on the flop can lead to a long term loss, if there is additional betting past the flop. However, the is a concept called Implied Value that is able to help particularly flush draws and open-ended straight draws remain profitable even with seemingly 'bad' odds.
Implied Value is a concept that takes into account future betting, and is most often used to anticipate your opponent calling on the river. If you have a flush draw, but are being offered pot odds of only 3 to 1 on the turn, if you have followed the above sections you will know that the pot odds are too low; you need at least 4 to 1 to make it a profitable call. However, this is where implied value can come into play. If you anticipate that your opponent will call you on the river if you do hit your flush draw, then although you are currently only getting 3 to 1 pot odds, you are anticipating 4 to 1 pot odds, and so are able to make the call.
Implied value also commonly plays a part where you are early to act in a betting round. Whilst the pot odds at the time you act may not be sufficient for you to call, if you anticipate callers behind you, this may swing the pot odds in your favour.