Poker Odds 101
Tuesday, 16. February 2010

All things being equal, poker is a game of odds. What are odds? If you make a bet and you are offered five to one or 5:1 on the bet, this means that if you win, the other person pays you 5 and if you lose then you pay the other person 1. For example, if you are at the dog track and you place a $10 bet on a dog paying 5:1; if you win then you collect $50 (plus the $10 that you placed on the bet) and if you lose then your $10 remains with the bookmaker. If you believe that the dog in question has better odds of winning, in this case better than 1:5, then a 5:1 bet on the dog is a good bet.

How does this translate to poker? Let us say for example that a player offers you 5:1 after the river card is dealt in Texas Hold’em and you think that you have a better than 1:5 chance of winning the pot, then you should call the bet. In most cases, it is difficult for you to be sure whether your hand is better than your opponent’s, but you should have some indication as to how likely it is that you are holding the best hand. In most cases, it’s a judgment call.

Example 1:
Let us take a real life example: You are in last position in a game of Texas Hold’em and you are holding [Q♥][8♥] on the turn and the board shows [A♥][K♥][6♠][2♦]. The player before you makes a bet. You know the player to be fairly conservative and most likely holding at least a pair of kings and more likely a pair of aces. In this case, you can assume that your only chance of taking the pot is if the river drops a heart which would give you the flush. As there a total of 13 heart cards in a deck and you have seen 4 of them then you are left with 9 outs or, in other words, there are 9 cards that could fall on the river that would give you the pot. What are your odds to win? Well, there are 52 cards in a deck and six of them are known to you at the turn; this means that there are a total of 46 cards (52-6) left out of the original 52 cards that could possibly fall on the turn. The fact that there are 9 cards left that could give you the win means that there are 37 cards (46 cards left unseen-9cards that could give you the win) that would make you lose. Therefore, you have 9:37, or approximately 1:4, chance to win the pot. If there is more than four times as much in the pot than it would cost you to call then you should call the bet or, in other words, if the pot odds are greater than 4:1 then you should call the bet.

Example 2:
Let us look at another example: Again, you are in last position in a game of Texas Hold’em but this time you are holding [9♠][8♠] on the turn and the board shows [8♣][7♠][7♦][6♥]. You know your player well and assume that he is holding pocket queens or higher. Where do you stand? Well, a ten will give you a ten-high straight so the four remaining tens are good for you (4 outs). Any of the four remaining fives would give you a nine high straight, so those are good too (another 4 outs). A nine would give you two pair (a pair of nines and a pair of eights), but that won’t help you if your opponent is holding an over pair, as you suspect, so forget that. An eight; however, would give you a full house which would be just fine (another 2 outs). Now, it’s time for the math:

4+4+2=10 cards that would win the pot
As in our first example, there are 6 cards in the deck that we have seen, leaving (52-6) 46 cards that we have not seen.
46(cards we have not seen)-10(winning cards)=36 (losing cards)

This means that the odds of you winning the hand are 36:10 against you winning the pot or 3.6:1. Now, let us say that there is $100 in the pot and your opponent decides to place a bet of $50. What are the pot odds? Back to the math:

$100(originally in the pot)+$50(opponents bet)= $150 total in the pot
$150(total in the pot)/$50(to call):1 or 3:1

In this case, with the odds of you winning the hand at 3.6:1 against you greater than the pot odds, 3:1 it would not be prudent for you to make the call.

It is a mistaken assumption made by many players that it is wrong to continue betting in situations where you are not favored to win. While this is true in most instances, it does not take into consideration any circumstances other than the current strength of your hand and what you believe your opponent is holding; it does not take into consideration the pot odds. Looking at Example 1 with the flush draw, if your pot odds are greater than the 4:1 odds against your winning then you should most definitely make the call. The lesson is simple: while you might find yourself losing more often then not when taking into consideration the pot odds, you will earn more money, statistically, in the long run.

A little confused? Let’s leave the world of poker for a second and simplify things and say that you are betting on the rolling of a die. You bet $1 that the die will come up 4. How much would you require in order to reap a profit from the bet? Well, there is 1 way you can win and 5 ways that you can lose. This means that you would require pot odds that are better for you than 5:1 in order to profit. In other words, if you were paid $6 every time you won and paid out $1 each time you lost then your pot odds would be 6:1. The pot odds of 6:1 are greater than the odds against you of winning, 5:1, so it would pay for you to take the bet. Let us say, for example you take this bet for 100 rolls of the die. The laws of statistics say that out of 100 rolls of the die you would win 20 rolls of the die (1/5 of the time) and lose 80 rolls of the die (4/5 of the time). This means that overall you would win $120 (20X$6) and lose $80 (80X$1) for an overall profit of $40.

Implied Odds
The theory of implied odds is quite simple and basically builds on what is written above. Let us go back to the situation outlined in Example 1 where you are chasing the flush and you are quite sure that your opponent is holding at least a pair of queens. Should luck be on your side and you do in fact manage to draw the flush on the river then you will, more likely than not, be able to hit him up for a big bet on the river if he has a pair of queens. In the event that he does not have a pair of queens and has two pair or, better yet, a set; then you should be able to empty his pockets. That is the idea behind implied odds. You sometimes should call even when the pot odds aren’t sufficiently large to justify it because there is the chance for a large pay out in the event that you do.

When it comes to No Limit poker, implied odds is the bottom line. In Limit poker, a powerful hand can certainly win you some chips whereas in No Limit poker, a powerful hand can win you the whole kit and caboodle.

While the theory of implied odds is incredibly important, you do not want to go bonkers. It certainly requires a modicum of moderation.

Calculating The Odds
Calculating the pot odds is not a problem. The size of the pot is a known parameter. If you are playing online then it is posted and if you are playing a live game then you can either count the chips or ask the dealer. The pot odds are simply the size of the pot in relation to the amount that you have to call. A simple example is if the pot is $10 and your opponent makes a $10 bet (bets the pot) then the total pot is $20. As you would be required to pay $10 in order to call your opponents bet, the pot odds are 20:10 or simply 2:1.

Calculating your odds of winning is the problem. There are only two cases in which you can be completely certain of your odds of winning. The first being when you are drawing the nuts. The second, and least likely of the two, is when you have seen your opponent’s hole cards. Otherwise, you are required to have some kind of read on your opponent or simply guess. For the most part, accuracy in calculating your odds comes down to knowing your opponents and good old fashioned experience.

Realism And Your Odds
In cases when you are drawing to anything other than the nuts (the best possible hand), there is always the risk, however slight it might be, that you still will not have the best hand. Drawing a great hand with your opponent drawing a better one is a recipe for destruction. When this happens, not only could you lose the cost of making a call but most likely lose quite a bit more in the event that you incorrectly believe that you have drawn the best hand. When the risk is miniscule, for example when drawing the king high flush, you can most likely ignore the risk. However, when you are drawing to overcards, for example, a certain level of suspicion is certainly healthy. In the end it comes down to your judgment; your knowledge of your opponents play and experience.

Common Poker Odds
Calculating your odds of winning is simply a matter of counting your outs. When drawing to the nuts, calculating your outs is a no brainer; otherwise, there is always an inherent risk. Below, I have listed some common poker odds all of which are based on certain assumptions which are laid out. The odds displayed below all assume that you are on the turn, 4 community cards are displayed, and apply to your chances of hitting the river. While many people want to know their odds of hitting two cards, e.g. from the flop, these odds aren’t incredibly helpful. If you need two cards to hit the straight or two cards to hit the flush, for example, you might want to reconsider making any calls. That having been said, you may certainly use a certain degree of extrapolation to determine those chance with the odds below in your arsenal.

Four To A Flush – 4.1:1
The assumption that we make here is that you are drawing to the nut flush. This is of particular importance if your hole cards are not suited and you are drawing to the flush. If your hole cards are suited and there are two or more cards of your suit present on the board then you can be fairly comfortable that you are holding the nuts as it is unlikely that you will be up against an opponent two hole cards of the same suit. If your hole cards are not suited and you are drawing to the flush then you should be very careful if you are not holding the ace or at least a high card.


Open Ended Straight Draw – 4.8:1
Let us say that you are holding [7][6] and the board shows [A][8][5][2]. You have 8 outs; the four fours and the four tens, all of which will give you the nuts presuming that there is no possibility of a flush. Now, if the board showed [A][K][9][8][2] then the tens would not give you the nuts as there is the possibility that your opponent is holding [Q][J] which would give him the higher straight.

Overcards With A Ragged Flop – 6.7:1
Let us say that you are holding [A][K] and the board shows [J][8][3] [2]and you suspect that your opponent is sitting on a pair of jacks without an ace or king kicker. You have six outs (any ace or king will give you a higher pair). The odds of 6.7:1 only holds if your suspicions as to what your opponent is holding are correct. This is a very risky assumption.

Holding A Pair And Drawing To Two Pair Or Trips – 8.2:1
Let us say that your holding [10][9] and the board shows [A][10][7][3] and you suspect that your opponent is sitting on a pair of aces. You have five outs to beat him; three nines (giving you two pair) and two tens (giving you trip tens). Now, your odds in this instance are based on the assumption that your opponent doesn’t have [A][10] or [A][9]. This can be a very dangerous assumption to make and you should have pot odds a bit higher than 8.2:1 to profitably make this call in order to make up for the times that you are actually drawing to half as many odds than you might suspect.

Inside Straight Draw (Gutshot Straight Draw) – 10.5:1
Again, we are making the assumption here that you are drawing to the nuts. Let us say that you are holding [7][6] and the board shows [K][Q][8][4]. Any of the four fives will give you the nuts as long as there is no possibility of someone drawing a flush. In the event that you are not using both of your hole cards to make the straight, for example let us say that you are holding [8][3] and the board shows [A][10][7][6]. A river [9] would give you a straight but not the nut straight as there is the possibility that your opponent is holding [J][8] which would give them the nut straight.

Drawing To A Set – 22:1
There aren’t too many circumstances that would warrant attempting to draw to a set, as you might imagine with 22:1 odds. Let us say that you are holding [6][6] and the board shows [A][Q][8][3]. Only one of the two remaining sixes will extricate you assuming, of course that no one else is sitting on a higher set.

Generic Formula
If you are looking to draw on something other than what is listed above then there is a fairly simple formula to figure out your odds:

((46-X)/X):1, where X=The number of outs

Conclusion
This is a very basic explanation of the mathematics behind calculating odds in poker; barely scratching the surface. That having been said, it should certainly serve to give you an idea as to what is going on behind the cards and a firm basis on which to build.

For further research into the topic, visit Full Tilt Poker. The Full Tilt Poker Academy offers a wealth of in-depth articles covering odds and poker strategies.

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