Applying pot and win odds at the poker table is essential to making good poker decisions. Some experienced players can do this simply by their excellent feel and judgment for the
game, but most of us need to rely on mathematics to help guide us with close decisions. Players who understand how to apply odds in a poker game will have a significant advantage over most of their opponents, Let’s look at an example to demonstrate how to apply odds to make good poker decisions. You are playing in a $1-$2 game and are seated on the button. You hold K♥ T♥ and the board shows 9♥ 2♣ 4♦ A♥. An opponent bets $2 for a total pot of
$10. To simplify the example, we assume your opponent is betting a pair of aces
but will fold if another heart comes on the river. Should you call to try and improve
to a flush?
Saturday, October 25, 2008
Mathematics of Poker | An introduction to Probability, Outs and Odds
words, you will improve to a flush once every five tries. If you played this hand five times, you should expect to lose $2 four times and win $10; once for a total profit of $2; therefore, calling has a positive expectation. What if the pot is only $7? In this case, you would expect to lose $2.00 four times and win $7 once for a loss of $1, so you should fold. To better understand this process, we begin the chapter with some definitions and
then discuss how to calculate various odds and probabilities for the most typical situations in Hold’em. The actual calculation however is not as important as how you apply odds to make better decisions, so we will discuss this process in detail.
Definitions
Odds and probabilities are two ways to express the same thing. Probability tells you how many times an event will happen. For example, you will I dealt a pocket pair once every 17 hands or 5.88% of the time. Odds tell you how many times an event will not happen. For example, the odds are 16 to 1 against being dealt a pocket pair.
Pot odds are the relationship between the current pot to the current bet. For example, if the pot is $100 and you must bet $10, the pot odds are 10 to implied pot odds is the relationship between the current pot and the bets you expect to win, to the current bet. Let’s look at an example of implied pot odds. You are playing in a $1-$2 same and your lone opponent bets out $2 on the turn. There is $10 in the pot, so your pot odds are 5 to 1; however, if you improve your hand on the river, you expect to earn at least one more bet from your opponent. You are risking $2 on the turn to win a total of $12, the $10 in the current pot plus your opponent’s $2 bet on the river; therefore, your implied odds are 6 to 1. If you expect that your opponent will bet out on the river and call a raise should you improve, you would earn two more bets, so your implied odds would be 7 to 1.
card that improves your hand. For example, when you hold two hearts and there are two hearts on the board, you need one more heart for a flush. There are nine remaining hearts or "outs" to improve your hand. If you have A♥ T♥ and you think another ace would also win the hand, you now have 12 outs: the nine hearts and the three remaining aces.
An out is counterfeited when a card that improves your hand gives an opponent an
even better hand. One of the most common mistakes made by many players is
assuming that they will win when a particular card improves their hand; however, it does you no good to draw to a hand that will only lose. For example, you could be hoping for a flush card only to lose to a higher flush or maybe even a full house. You could hit an overcard, a card higher than any card on the board, only to lose to two pair, three of a kind, a straight, or a flush.
When applying odds, you should discount an out whenever there is a chance that you could improve but still lose the hand. Once you know the number of discounted outs that can win the hand, you can calculate the odds against improving to the winning hand to determine your best strategy. How much you discount an out is dependent on how many players you are against and you read on your opponents’ possible holdings given the betting sequences in the hand. For example, you have three outs to an overcard ace and feel that you might win
about 2/3 of the time against a lone opponent if you hit the ace; therefore, you would discount your three outs to two outs. However, against two opponents you might feel you will only win about 1/3 of the time, so you discount your three outs to one out. If you are against three or more opponents, you might feel that even with another ace, there is a high chance that you will not be able to win the pot. In this case, you should disregard the outs to the ace since you are drawing dead. Drawing dead is when you cannot improve to the winning hand. This occurs when your opponents counterfeit all of your outs or already have a hand better than the one you are drawing to. For example, you might be drawing dead to two overcards if an opponent already has three of a kind, two pair, or your outs would give your
opponent an even better hand. We will go through several examples to look at how you should determine the number of discounted outs you have in a hand based on the probability that your outs are counterfeited or that you are drawing dead. First let’s look a how to
calculate odds.
Calculating Odds
To determine the odds against improving your hand on the next card, compare the total number of cards that will not help you to the number of cards or "outs" that will. For example, you hold 7♥ 6♥ with a flop of A♣ T♥ 5♥. On the flop there are 47 unseen cards. Out of these 47, there are nine hearts remaining that will improve your hand to a flush and 38 cards that won’t; therefore, the odds against improving to a flush are 4.2 to 1 (38/9). An open-ended straight draw has eight outs, which is 4.9 to 1 against improving (39/8). An inside straight draw, a.k.a. gut-shot draw, has four outs, which is 10.75 to 1 (43/4). If you don’t improve on the turn and want to know the odds that the river can will improve your hand, the odds will improve just slightly as one more care has been seen. There are only 46 unseen cards on the turn; therefore, a flush draw is now 4.1 to 1 (37/9), which is just slightly better than the 4.2 to 1 odds you had when drawing on the flop.
To determine the probability of improving on the next card, simply divide your outs by the total number of cards left in the deck. For example, the probability of improving to a flush on the next card is 19% (9/47). You will improve to an openended straight 17% of the time (8/47), and a gut-shot straight 8.5% of the time (4/47). I prefer to know the odds are 11 to 1 rather than the probability is 8.5%, because it is easier to compare to the pot odds you are receiving. Sometimes on the flop, you want to know the probability that either the turn or the
river card will improve your hand with two cards to come. These calculations are slightly more complicated. The best way is to multiply the probability of missing on the turn by the probability of missing on the river. For example, for a flush draw you would multiply 38/47 by 37/46, which equals 1406/2162 or .6503; therefore, 65% of the time you will not improve and 35% of the lime you will. [To convert this to odds, invert the percentage and subtract 1 to get 1/.35 -1 = 1.9 to 1 against improving. This section looked briefly at how to calculate simple odds and probabilities; however, calculating odds in your head during a poker game can be quite cumbersome. In reality, all you need to do is memorize the following chart.
Determining the Number of Discounted Outs
When calculating odds, you need to use the number of discounted outs that will help you win the hand. As discussed before, it does you no good to improve your hand only to lose to a better hand. Let’s look at some examples to see better how you determine the number of discounted outs. You have K♦ Q♣ and the board is J♦ T♣ 5♥ 2. You have eight strong outs to the nut straight with any ace or 9 and six weak outs to the king or queen. The six outs to the king or queen are weak since your opponent could ahead;
To have two pair or a set or is counterfeiting your outs. In this example, a king would give you a pair but might also give an opponent a straight, two pair, or a pair with a better kicker. Note all the hands you would lose to if a king comes: KK, JJ, TT, 55, 22, AK, AQ. KJ, KT, K5, K2, Q9, JT, J5, J2, T5, and T2. If a queen comes, you would lose to QQ, JJ, TT, 55, 22, AK, AQ, K9, QJ, QT, Q5, Q2, JT, J5, J2, T5, T2, and 98. How much you should discount your weak outs often depends on how many opponents you are against. In the example above, you have six weak outs. Against a lone opponent, if you feel that 50% of the time a king or queen will help you win, you should discount the six weak outs to three. In this case, you would play the hand as if you had an equivalent of 11 outs to win the hand, the three discounted outs and the eight strong outs to the nut straight. If you are against two opponents, you might estimate that a king or queen would win only once every six times; therefore, you would play as if you had nine outs, eight nut outs to the straight plus the one discounted out. Against three opponents, you should probably disregard the weak outs since it is unlikely a king or queen will win. In this case, you would play only if you draw to your eight nut outs is justified. Let’s look at some more examples. You have A♣ T♥ and the flop is K♦ T♣ 5♠. You have two strong to the ten, unless an opponent holds KT or T5. Another ace would give you two pair, but your out is counterfeited if an opponent holds AA, AK, or QJ, so you should discount the out to the ace. All your outs should be discounted slightly for the possibility that an opponent holds a set. Depending on the number of opponents and the betting sequences, you should play this hand as if you had between two and four outs. You have A♣ 9♥ and the flop is J♦ 9♦ 4♣ with several callers on the flop. You probably are against a flush draw, so the A♦ is counterfeited. You could also lose to another ace if someone has AA or AJ. Always account for the possibility of a set.
Advanced Concept: Whenever the flop is two-suited, you should discount a suited out against a lone opponent and probably disregard the out against several opponents for the risk that one of them holds a flush draw. A common mistake made by many players is drawing to weak hands when flush draws are likely. As a general rule, most draws are not profitable with a two-suited flop arid several callers in the hand. The only exception to this is when the pot is
exceptionally large. This is a key concept since you will be playing with a two- or three-suited flop about 60% of the time! This concept is discussed further in the flop chapters. For now, simply understand that you need to discount or disregard your outs based on the likelihood that they are counterfeited. Another consideration when determining your outs on the flop is the possibility that you could improve on the turn only to see an opponent improve to an even better hand on the river. Advanced Concept: When drawing on the flop, you should discount your outs a little, and maybe a lot, for the probability that your opponents could draw to an even better hand on the river. There are very few hands that are a lock to win on the turn. Nut flushes can lose to a full house if the board pairs on the river. The nut straight can lose to a flush on the river. Your two pair could lose to an opponent hitting a set. When the flop is twosuited, these types of situations occur often since there are a lot of river cards that could hurt your hand.
Most players complain about their bad luck when they improve on the turn to lose
on the river. Good players recognize that these types of situations occur a lot and
include this possibility in their decision-making process. Borderline draws on the
flop should often be folded for the possibility that you will lose on the river.
Now that we know how to determine the number of discounted outs and calculate
the odds against improving to the best hand, we can look at how to apply odds at
the poker table.
Application of Odds
The basic steps in applying odds at the poker table are as follows:
1. Determine the number of discounted outs.
2. Calculate the pot odds. This is the size of the pot in relation to the bet.
3. Calculate the implied pot odds. This is the current pot plus the bets you
expect to win in relation to the current bet.
4. Compare the implied pot odds to the odds against improving your hand
5. Determine your best strategy.
rock who never bluffs, bets out and the middle player folds. What should you do? • Determine the number of discounted outs. We assume your opponent has at least a pair since he never bluffs; therefore, you need a king or queen to
improve, which is six outs. You would be drawing dead against TT, 77, or 55, unless you hit a runner-runner straight. Other likely holdings of your opponent include AT, KT, QT. and JT. In this case, a king or queen would
not help against either KT or QT. It is doubtful that your opponent would call a raise preflop with K7, K5, Q7, Q5, T7, 75, or T5: therefore, you only need to discount your outs for the probability that your opponent holds KT, QT, TT,
77, or 55. One other consideration is what could happen if you hit the king or queen on the turn. Your opponent could possibly win on the river by hitting two pair or better. You should discount your outs a little more for this possibility. To determine how much you should discount your outs, it is helpful to evaluate the probable hands of your opponent. Probable hands that you could beat if you improve include JJ, AT, A7, A5, JTs, and 99. Discounting
outs is always a matter of judgment, but you might expect to win this hand 50% of the time when you improve, considering the possibility that your opponent might have a set, KT, QT, or improve on the river. Therefore, you should discount your six outs and play as if you had three outs.
• Calculate the pot odds. The total pot at this point is $75 (three players paid $20 to see the flop + $5 small blind + $10 bet on flop by the big blind): therefore, your pot odds are 7.5 to 1 for a $10 her.
• Calculate the implied pot odds. Do you expect to win more bets when the king or queen comes? You should win bets 50% of the time when you improve, but you will lose more bets the other 50% when your opponent has a better hand. A simplified assumption would be that all future bets break even.
• Compare the implied pot odds to the odds against improving your hand. In this case, we look at the pot odds since the implied odds are the same. The pot odds of 7.5 to 1 are compared to the odds against improving with three outs of 15 to 1 (see out chart).
• Determine your best strategy. The odds against improving are 15 to 1; therefore, we should fold since the pot odds are only offering 7.5 to 1. Let’s discuss this hand a little further to show the importance of discounting outs. Many players draw to overcards on the flop hoping to pair up, and this example shows that this often is a big mistake. If we played our hand thinking we have six outs to the king or queen, our odds are 7 to 1 against improving. This compares favorably to the 7.5 to 1 pot odds; therefore, we would call expecting to make a
small profit. However, this assumes we would always win when the king or queen comes. As we discussed before, our opponent could very well have KT, QT, TT, 77, 55 or beat us on the river. Some players also justify calling by saying that they have implied odds of winning
more bets should they improve. This is true if your hand wins, although sometimes
you won’t even collect more bets when your opponent folds on the turn to a bet or
raise. The problem is that sometimes you will lose additional bets. If your king or
queen comes on the turn, you will probably raise and then be faced with a reraise, if
your opponent has a set or two pair. Let’s look at another example of $10-$20. An early and middle position player call. You call on the button with A♣ 5♣. The small blind calls and five players see the flop of K♣ 9♣ 4♦. The small blind bets and the big blind folds. A strong player in early position raises. The middle position player folds. What should you do?
• Determine the number of discounted outs. The early position player most likely has a pair of kings and might have 99. The small blind most likely has a pair of kings, K9, 99, 44, or possibly a flush draw. You have nine outs to the nut flush and three ours to the ace. If one of your opponents has a set or two pair, you could hit your flush and possibly lose to a full house; therefore, a small discount I needed. An estimate might be to discount your flush draw from nine outs to eight outs. Your three outs to the ace need to be discounted since you would lose to AA, KK, 99, 44, AK, A9, A4, K9, K4, 94, and for the possibility that someone hits a better hand on the river. Again, (his is a matter of judgment, but you might estimate that a pair of aces would win about 33% of the time; therefore, you could discount your three outs to one out. As a result, I would play the hand as if J had nine discounted outs.
• Calculate the pot odds. The total pot at this point is $80 (five players paid $10 to see the flop + $10 bet on flop by the small blind + $20 raise by the early position player). You face a bet of $20, so your pot odds are 4 to 1.
• Calculate the implied pot odds. If you hit the flush on the turn or river, you can expect to gain some extra bets, especially if one of the players has a set. Since there are two opponents in this hand, you might expect to gain at least one big bet on the turn and one big bet on the river for a total of $120 ($80 +$20 + $20). Your implied odds are 6 to 1 faced with a $20 bet. Note: A big bet is the amount of a bet on the turn and river, compared to small bets on the first two rounds of betting.
• Compare the implied pot odds to the odds against improving your hand. Nine outs are 4 to 1 against improving, which are equal to the pot odds of 4 to 1; however, your odds compare favorably to the implied pot odds of 6 to 1.
• Determine your best strategy. Calling is profitable. Raising is a consideration to try to buy a free card (see "Deceptive Tactics" chapter). Let’s look at one more example of $10-$20. You raise in early position with J♥ J♠. Two middle players, the button, small blind, and big blind all call for a total of six players. The flop is T♣ 8♦ 8♥. It is checked to you, and you bet. One middle player, the button, and small blind call. Four players see the turn card of Q♦. The small
blind checks and you bet. The middle position player raises and everybody folds to you. There is $220 in the pot. What do you do?
• Determine the number of discounted outs. Assuming the middle player is not a tricky opponent, your opponent has at least a pair of queens with a band like AQ or KQ. He might also have TT, 88, or A8. QQ is unlikely since he probably would have reraised preflop. Q8, J9, and T8 are unlikely since he probably would have folded to a raise before the flop. You have four outs 10 a straight and two outs to a full house. Your two outs to the full house are
strong since the only two hands that would beat you are QQ and 88. Your four outs to the straight are relatively strong unless your opponent has QQ, TT, 88, or 98, QQ and 88 are unlikely, but TT is a decent possibility. Only a weak player would call a raise preflop with 98s. One other small possibility is that your opponent has QJ, in which case you would split the pot if a 9 comes. Therefore, I would only discount your six outs by one out to account
for QQ, TT, 88, and QJ, and play the hand as if you had five outs.
• Calculate the pot odds. The total pot is $220 and the bet is $20, so your pot
odds are 11 to 1.
• Calculate the implied pot odds. You should expect to earn another bet on the river if you improve. You might lose two bets on the river if you come out betting with the straight and lose to a full house. You might estimate that you would win $15 on average when improving; therefore, the implied odds are $235/20, which are 11.75 to 1.
• Compare the implied pot odds to the odds against improving your hand. 11.75 to 1 implied pot odds compares favorably to the 8 to 1 odds against improving with five outs.
• Determine your best strategy. Given the large pot, calling is correct. Let’s change the scenario in the above example to look at the effect of a large pot. Let’s assume you raise preflop as before and the middle position player calls, but now everyone else folds. The flop and turn are the same, and he raises once again when the Q♦ comes. You do not believe your opponent is bluffing. With only $135 in the pot and the $15 you expect to earn on average should you improve, your implied pot odds are now only 7.5 to 1. This is less than the 8 to 1 odds against improving, so you should fold.
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