Monday, May 17, 2010

Back to Basics: Pot Odds, Hand Equity and Implied Odds

I'm starting to realize that some of my readers are quite new to poker... or perhaps don't have basics and fundamentals down pat.  In addition to my random strategy posts, I plan on starting a new segment call "Back to Basics."  To most readers, this will be elementary, but I feel it serves good purpose to hammer down fundamentals of pre- and post- flop play.  Perhaps I will learn a thing or two from the comments section, as well.  I can't promise anything, but I do plan on trying to post a new "Back to Basics" every Monday.

As a quick recap of prior posts, readers interested in this series would be best served by reading my thoughts on Poker Tracking / Datamining software, Bankroll Management, Rakeback Agreements, and Pre- / Post- Game Process Parts I, II, III & IV, to have a good footing on terms, assumptions and an understanding of my poker philosophy.  Also see prior "Back To Basics" back-posts.

Welcome back to the second segment of Back to Basics.  This entry will cover two fundamental poker formulas: pot odds and hand equity.  The two term, for the most part, go hand-in-hand, for the simple fact that a player can comprehend the formulas and instantly apply them to his or her own game.  It should be noted that a player not using these two ideas who begins to apply them to their game is a better player than before, but these ideas should not be used in a vacuum; i.e. as a good player, you need to consider your opponent's holdings vs. your own.

That said, let's get into the definition of both terms.

Pot Odds:
As defined by wikipedia (the internet gospel on *EVERYTHING* :-) ) it is "... the ratio of the current size of the pot to the cost of a contemplated call. In other words, if the pot contains $100, and a player must call $10 to stay in the hand, then the player has 100-to-10, or 10-to-1 (commonly expressed as 10:1), pot odds. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value. Indeed, a common usage of the term is to say that one "has pot odds", meaning that the present pot odds, compared to one's estimated chance of winning, make it profitable to call."

Personally, I am terrible with ratios, so I like to convert these ratios to percentages, since I am a lot better equipped to handle one simple number... therefore I add the two numbers together and divide by the first (in my head, so I give a rough estimate of the number).  For the example above, to demonstrate my thought process, I add 10 + 1 (=11), take the 1 from the ration and divide it by the sum 11.  I know that 1 into 11 does not go evenly, but 1 into 10 does (=.1 or 10%), so I'm going to rough the number by saying it equals about 8-9%; slightly less than 1 into 10...  It's a simple shortcut, but I don't need to be exact since poker is not a game of exact numbers. (You're making educated guesses at your opponents cards anyway, right?)

What do pot odds tell us in layman's terms?  They tell us the price we're being offered to act upon (fold, call, raise).  I don't know about you, but I want a bargain when it comes to purchases; I don't want to overpay for things that I can get cheaper.  The same holds true with poker; if my opponent is offering me a bargain, I take it.  If not, I fold.

What?  I don't understand!  Who cares?  Allow me to illustrate the concept in terms of flipping a 2-sided coin: for every coin flip of heads you pay me $6 and for every flip tails, I pay you $4, you would be getting (in this case straight, not pot, because we're talking about coin flips, not cards) 4-to-10 on your bet, or 40%, on what we all know to be a 50/50 (50%) true odds bet.  I'm giving you an awful bet, and you should [rightfully] say "Go pound sand," or something less polite :-).  Perhaps in the short term, 1, 2, 5, even 25 flips, you may hit more heads than tails and actually make money.  In the long term, though, you know you're going to lose money - flip that coin every day for the rest of your life and you'll be a big loser.  However, if the situation is reversed, I'd become your best friend because I'd essentially be giving money away.

Implied Odds:
Again, courtesy of wikipedia (the internet gospel on *EVERYTHING* :-) ), implied odds "... are calculated the same way as pot odds, but take into consideration estimated future betting. Implied odds are calculated in situations where the player expects to fold in the following round if the draw is missed, thereby losing no additional bets, but expects to gain additional bets when the draw is made. Since the player expects to always gain additional bets in later rounds when the draw is made, and never lose any additional bets when the draw is missed, the extra bets that the player expects to gain, excluding his own, can fairly be added to the current size of the pot. This adjusted pot value is known as the implied pot.

Example (Texas Hold'em)
On the second to last betting round, Alice's hand is certainly behind and she faces a $1 call to win a $10 pot against a single opponent. There are four cards remaining in the deck that make her hand a certain winner. Her odds of drawing to one of those cards is 10.5:1 (8.7 percent). Since the pot lays 10:1, Alice will lose money by calling if there is no future betting. However, she expects her opponent to call her additional $1 bet which she will make when she makes her draw. She will fold when she misses her draw (and lose no additional bets). Her implied pot odds are 11:1 ($10 plus the expected $1 call, to her additional $1 bet). This call now has a positive expectation."

Simply put: What are the odds that I will get the rest of my opponents stack by making the play in question.  How does all of this tie into Hand Equity?

Hand Equity:
Every hand holds a particular value.  Each hand has a different value, or chance of winning vs. a different hand (or set of hands).  If everyone were dealt hands face up, you could compare two hands against one-another and draw certain mathematical facts.  It is from these hand values that you can conclude your percentage chances of wining a particular hand.  Just as in the coin flip example above, where you know if you flip a coin an infinite number of times, you will be exactly 50/50 on your heads and tails, if you were to deal a combination of cards against one another an infinite number of times, (although there are many more variables) the particular hand in question vs. the opposing hand is going to win a certain percentage of the time.

  • From a pre-flop perspective, there are hand rankings where the strongest hands have more equity vs. the weakest hands.  For example, AA, the strongest starting hand in Hold'em, has ~80% equity when seeing a flop heads-up vs. KK (the second strongest starting hand in poker), which is true for all pairs vs. lower pairs all the way down to 33 vs. 22.  On the same vein, any pair vs. any two overcards (e.g. 88 vs. KJ or 22 vs. AK is worth roughly 50% equity.  If you are interested in running a Monte Carlo simulation on hands vs. other hands, there is a GREAT free tool available, called PokerStove, which allows you to simply plug in 2 or more hands and compare them against each other.  Since there are 1326 possible starting hand combinations in Hold'em (just trust me, I'm not going to do the math), the pre-flop hand comparisons are fairly static in nature.  In other words, over time, you will be able to memorize the average strength of hands against other hands, which is why pre-flop hand selection is so critical to a winning player.  For more information on starting hand strength, see my post Odds Scenarios.  It may be a tremendous help the fledgling player.

    Realize as I'm sure you have by now, that any two cards can win a hand.  You can play 72o (which is the theoretically worst starting hand combination in hold'em) against AA and win at a clip of 12.5% of the time if both players see the hand through to the river. Will that win you money over the long term?  Perhaps, if your opponent is dumb not intelligent and you can get all of the money in the center when you're ahead (in the 12.5% of the time), and are able to fold the other 87.5%.  You have the implied odds to play just about any two cards against him, even though you are not getting appropriate pot odds.  However, the real world is not a vacuum, and not all players hold AA to your 72o, nor are they always willing to push all-in on your 72o against a board of 7-7-2 or A 3 4 5 x.  Therefore, your hand selection is important, because 72o, when played over the long run, is a money loser.

    Tying directly into pot odds, though, let's say UTG (who is an extremely tight player who would only raise holding AA) raises 3x BB on a 9-person table.  You sit in the big blind holding 72o, and watch as the entire 7 players preceding him call.  The pot would contain 9 (players) * 3x (big blinds) + 1 (for your big blind) 28 total big blinds.  You'd be getting 28 to 3 (by my math, roughly 9%) pot odds to make the call with literally ANY TWO CARDS; it'd be well worth it to you to make the call in that spot.
  • From a flop and beyond perspective, though, you can now calculate your direct equity from the hand you hold.  Since you can now complete a 5 card hand, you can compare your 5 card hand and, if you feel it is an underdog, calculate the odds of improving to the winning hand.

    At the heart of this process of odds calculation is your ability to count the number of cards which will improve your hands by the turn or river.  You need to be able to somewhat accurately put a hand to your opponent.  For example, if you are holding AK, and a flop of 5 7 T hits, you need an A or K on the turn or river to potentially improve your hand to the winner (assuming your opponent has a pair of Queens or lower.

    To continue with the example above, you can calculate your available out cards as the 3 remaining Aces in the deck (4 total, you hold 1, therefore 3, in theory remain), and 3 remaining Kings (same logic as counting Aces), which gives you a total of 6 out cards to improve your hand.  Keeping with imperfect math, we want to get a rough idea of the percentage of time our hand will improve to the winner.  We can do this by multiplying the number of outs we have for improvement by 2x the number of cards we expect to receive on future streets.  Given the above example, we are on the flop and expect 2 more cards (turn and river).  Therefore, we multiply our 6 out cards x 2 (the magic number) x 2 cards expected to receive to get our estimated chances on having a winning hand by the river.  The product of 6x2x2 = 24%.  P.S.  If you've ever watched the World Series of Poker, or any other poker series (High Stakes Poker, etc.), you'll see percentage odds on the side of your screen for each hand in play.  Armed with the rule of 2x and 4x, you too can quickly (and roughly figure out the equity percentages of each hand.

    Note: Pertaining to the example above, you need to be sure that your opponent did not flop a set (pocket pair + 3rd of the rank on the board for 3-of-a-kind), because even if you turn an Ace and river and Ace, you are holding 3-of-a-kind to his rivered [Tens, for example] full of Aces for the rivered boat.  Following, you are almost drawing dead on the flop to a set.  Just the same, you need to carefully put your opponent an appropriate hand; i.e. if he holds AT to the 5 7 T board, only a King will improve your hand because a turned or rivered Ace will give him 2 pair and you one; the rule of 2x 4x would give you 3 outs, or 3 x 2 x 2 = 12% to the winning hand in that case.

    One more example: a popular one; flush draws:
    Referring to the previous post Back to Basics post on suited connectors, let's create a scenario where I hold 9s8s and the flop comes As 2s Jd.  Assuming that my opponent does not have a higher flush draw, but holds a pair of Aces or Jacks, I can count my out cards as follows: 13 total spades (as with every - 2 on the board - 2 in my hand leaves me with 9 spades.  I expect 2 cards to come, yielding me a hand equity of 9 (outs) x 2 (turn card) x 2 (river card) = roughly 36% equity in the hand.

    Now, let's change the flop around a big and give ourselves a flop of: 6s 7s Ad... we have a flush draw, a straight draw, and a straight flush draw.  At this point, for all intents and purposes, we have a MONSTER hand which holds at least 50% equity against almost all hands.  How?  We need to take care in counting our outs, because we don't want to count cards twice, a common mistake, but here goes:  We know we have 9 outs for the spade draw, as in the above scenario.  However, we also have an additional 4 5's and 4 Tens, which will give us a straight.  Ooops!  Are there really 4 5's and 4 Ten's though?  We've double-counted our ranked cards because we've already counted the 5s and the Ts in our spade out cards as 2 of the available 9.  Therefore, there are really only 3 5's and 3 Tens which we can cleanly count, giving us 6 straight out cards.  9 flush + 6 straight out cards gives us 15 total out cards x 2 (turn card) x 2 (river card) = 60%(!?!?!?!?!) equity.  How is that possible?  I'm actually ahead without having a made hand... which emphasizes the power of a suited connector (yes, you are ahead of a 1 pair made hand such as Aces or pocket pairs).  If you put your opponent on a pair, it would not be incorrect to get all of the money in the middle given any pot odds.  Look at the same example and figure out your equity if you put your opponent on a set of 6's or two pair: Aces and 7's .  What about if your opponent has the AsKs (answers at the bottom)?
Tying it all together
If you haven't already jumped to the conclusion, let me sum it up for you.  A simple / quick estimate of whether you're getting an immediate bargain is comparing your hand equity to the pot odds offered.  Simply, if the pot odds are less than the hand equity, then it's an easy call (if you're facing a 20% pot bet but your hand equity is 32%).

If the pot odds are not as good as the hand equity, but you feel as though making your hand would get your opponent to stack off his remaining stack to you, then mix in the element of implied odds (such as the example of playing 72o vs. AA).  You usually want to have a good amount of your villain's stack remaining on the river in order to get proper implied odds.

* Given you hold 9s8s and the flop is 6s7sAd:
  • If your opponent holds 6 6, your equity is reduced by one out, the As, which would give him 6's full of Aces vs. your Flush; 14 (outs) * 2 (turn) * 2 (river) = 56%.  Your equity is also reduced by another TBD (to be determined) card on the turn, 13 (outs) * 2 (river) = 26% chance of improving from the turn to the river.
  • If your opponent holds A 7, your equity is reduced by one out, the As, which would again, give him Aces full of 7's vs. your Flush.  However, you do not lose an out card on the turn because the As is the only card that will make your hand while betting his.
  • If your opponent holds AsKs, then you are in a whole new ballgame; the only out cards you have are any 5 and any Ten, making you either a straight or a straight flush.  Therefore 4 5's + 4 Tens = 8 * 2 * 2 = 32%.

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