Hold 'Em Mathology

Click the following titles to access the posts.

Click the Archives button near the top of the screen to see summaries of all posts with embedded links.

1. Welcome

2. Introduction to Expected Value- Definition, Part 1

3. Introduction to Expected Value-Hero Calling, Part 2

4. Introduction to Expected Value- Hero Betting, Part 3

5. Introduction to Expected Value- Math Summary, Part 4

6. EV Analysis Without Explicitly Assuming a Villain Range

7. Baseline Stack for EV Analysis

8. Using Combinations in Poker Calculations

9. Critical Values for Various Pot Odds

10. Thinning the Field. A Good Idea or Not?

11. Does First to be Dealt Have an Advantage?

12. Advanced Implied Odds Model, Part 1

13. Advanced Implied Odds Set Mining, Part 2

14. Advanced Implied Odds Flush Draw, Part 3

15. Advanced Implied Odds Gut-shot Straight Draw, Part 4

16. Advanced Implied Odds Flush and OESD Draw, Part 5

17. Advanced Implied Odds An Excel Application, Part 6

18. Advanced Implied Odds, Summary, Part 7

19. c-Bet: Hitting the Flop Probabilities

20. Multiway EV Analysis

21. Bet Sizing Holding the Absolute Nuts

22. A Quick Estimator For an EV-Based Call Decision

23. Button Steal With Zero Equity

24. A Check EV Example

25. Bet vs. Check When Check Equity Exceeds Bet Equity

26. Bankroll Management and Risk of Ruin (RoR)

27. EV Analysis When Fold Equity Is Greater Than 50%

28 Can More Than One Player Be +EV?

29. EV For Small Blind Opening Preflop

30. Small Blind Play – Open Bet Size and Shove Analyses

31. Small Blind Pre-flop Opening Raise for +EV

32. Quick Review- Estimating Required Equity For Calling an All-In Bet

33. On Estimating Card Equity. Part 1

34. On Estimating Card Equity, Part 2

35. Add a Math Analysis to Your Decision Checklist

36. When Is A Villain Fold Better Than A Call?

37. Bet Sizing. Two Case Re Fold Equity.

38. Button Pre-Flop Open Raises and Steals

39. ROI, ITM and Profit Statistics for STT SNG’s

40. Required Equity and Range for Calling an All-In Pre-Flop Shove

41. Over-cards to a Pair on the Flop, Turn River

42. Interaction of Board and Player Hands. Aces and Flush Examples

43. Bad Beats and Suck-Outs

44. Basic Bet Sizing Theory

45. Using An Equity Reduction Factor For Estimating Multiway Equity

46. With Equity > 50%, Determining Bet Size To Deny Villain Required Pot Odds

47.Early Street Analysis. Part 1 Estimating Future Bets

48 Early Street Analysis. Part 2 Aa Constant Bet Factor Model

49. GTO Strategy for a No Entry Cost Game

50. Minimum Card and Fold Equities For a Profitable Shove

51. Big Blind Strategy Against One Limper

52. The Cost of Folding

53. Modifying The Equal Capability Assumption

54.Getting a Worse Hand to Call

55. Probability Review: Aces, Who Has Aces?

56. Implied Odds: Analysis of Villain Call Probability

57. Equity of Various Hands Against Various Villain Types

58. Fold Equity Exploration

59. Equity and Value Bets

60. Playing Stake and Bankroll

61. Opponent NHLE Hand Probabilities

62. Simple Call/Raise Option Knowing Villain’s Pre-Flop Range

63. The Suit Interaction Effect for Pair vs. Pair

64. The Suit Interaction Effect for AK vs 22

65. Equilab is Great But Should Be Used Cautiously

66. The Fair Chance Equity Rule for Calling a Bet

67. A Typical Math Based Approach To Answering a Typical Hold’em Decision

Click the Archives button near the top of the screen to see summaries of all posts with embedded links.

Introduction. “When you know a villain is capable of pushing 15/20bb with low PP like 22-66, when is it OK to call them with Ax/Kx type of hands?”

The above is representative of many types of questions one sees on poker forums? If in situation X, villain does Y, what do I do holding Z? Naturally there is no one response for every X, Y and Z, or is there? In this article we think that there is, perhaps not for every X, Y and Z and probably not a response that is complete but maybe one that can serve as an initial cut to a decision. And that response is simple in theory but not always simple in execution. It is to estimate the expected value (EV) of the possible decisions and pick the one that provides the maximum. So, we’ll outline how one goes about it and illustrate it with addressing the above question that was actually asked on a poker forum.

A General EVcall Equation

Here is a general EV equation which can serve for many but not all poker decision analyses, namely a hero call. We have left out raises (as bluffs or for value) and other possibilities such as floating and implied odds.

EVcall = eq*(Pot +Bet + Call) - Call

where

EVcall is EV for a hero call

Pot = pot before villain’s bet

Bet = the amount of the bet

Call = the amount hero has to call

eq = the probability hero wins in a showdown

Application. Now to address the illustrative question. This is a hero call decision. No pot was given so we’ll assume it is 30 and we’ll also assume the bet is 20, which hero must call or fold. So, this is what we know:

Pot = 30 Bet = 20 Call = 20.

We don’t know the equity but we do know villain’s and hero’s ranges, 22-66 vs Ax, Kx. A poker equity calculator such as Equilab can help us in this regard. We’ll consider two possibilities for x, the kicker; x <= 6 and x > 6. The equity results are as follows assuming no pairing of the ace or king:

22-66 vs Ax, Kx for x <= 6: eq = 38%

22-66 vs Ax, Kx for x > 6: eq = 47%

Applying the EVcall equation, we have:

EVcall for x <= 6: 0.38*(30 +20 +20) – 20 = 6.6

EVcall for x > 6: 0.47*(30 +20 +20) – 20 = 12.9

In this case both sets of kickers provide positive EV. With other pot and/or bet sizes that might not be the case especially for the lower equity range.

Of course, you could do much more with the EV math. For example, suppose you wanted to know what the minimum pot size has to be to make good call with a lower kicker. The applicable EV equation is the following:

EVcall for x <= 6: 0.38*(Pot + 40) – 20 >= 0

Then, Pot >= (20- 0.38*40)/0.38 = 12.6

So, hero has +EV with any low Ax or Kx kicker to call a shove of 20 as long as the pot is greater than 12.6.

If hero was the bettor, a similar type of analysis can be done to determine such factors as critical equity or bet size.

Summary. Calling a bet is probably the easiest application of an EV math analysis, but is only exact if the call closes the action, such as with an all-in bet or a river closing call. If there are to be future bets, the result should only be considered as a first-cut look.

Introduction: Determining the equity you need to guide you towards a good poker decision is a common math challenge. It typically involves determining the equity value that maximizes your expected value or, at least, insures no loss. In this posting, I focus on calling a bet or folding, using what I call the Fair Chance Equity rule (FCE).

Situation: We will examine three heads-up situations, assuming the pot is 60 and villain bets 20, making the pot total to 80 before hero has to call or fold.

Case A: The bet is not all-in and the street is not the river.

Case B: The bet is not all-in but the street is the river

Case C: The bet is all-in

The Fair Chance Equity Rule: There are several ways to determine how much equity you need to math-justify calling a bet. I believe what I call the Fair Chance Equity rule is the most intuitive. For our sample case, after villain bet, the pot is 80 and you have to call or fold (we are ignoring raises as that leads to much complication). If you call, the pot increases to 100, all of which the winner collects. To justify the call, since you invested 20% of the total pot (20/100) at the decision time, a “fair return” would mean that 20% is the minimum frequency with which you should win, hence the minimum equity.[1]

Let’s now see how this works for our three cases:

Case A: Not All-in, Early Street. There is no reason the Fair Chance Equity rule cannot be used but it would only be exact if the hand were checked down. The reason, of course, is that betting is likely to take place on future streets, so that the calculation can only provide an initial cut at the proper decision. Also, there may be justification for calling even if you do not have the required equity such as floating and implied odds.

Case B: Not All-in and River Bet. Here a call closes the action so the FCE calculation provides an exact result. Clearly, a raise is a possibility and then the hero essentially becomes the bettor. If hero can estimate his equity, then using the same rule can provide insight on how much to bet, but we’ll leave that to another posting.

Case C: An All-in Bet. Here there can be no future betting so the FCE rule is exact in all respects.

Summary: The FCE rule is both simple and intuitive IMO. It provides an estimate of the equity you need to justify calling a bet. Of course, it’s most useful when you can reasonably estimate your equity and that is true for all EV math calculations. Clearly, any initial math-analysis should be adjusted for factors not explicitly considered such as position, stack sizes, villain characterization, etc.

Equilab is a popular and free equity calculator. You enter your hand or range along with hands or ranges for one or more opponents. The program will then provide your showdown equity, either through simulation or computation.[1] Not only that, you can also enter flop or flop and turn cards for an equity result.

So, with Equilab (and other similar equity calculators) providing this important factor, your winning chances, what is the concern?  It is that an equity calculator assumes the hand gets to showdown, which, of course, is not always true. On an early street, hands get to showdown if a bet is all in or all players check down. Hands don’t get to showdown if all but one player folds. And folding is not an uncommon action. Also, of course, estimating your opponent’s hand or range is subject to error, something which Equilab doesn’t directly consider but you can apply it over a range of inputs.

This leads us into the notion of equity realization, the chance that you will actually realize or even better the showdown equity. I would suggest that 33 has lower realized equity than 89s although it usually has higher showdown equity than 89s against a specific range. For example, 33 has showdown equity of 46% against a top 25% hand while 89s has a lower equity of 37%. But, if the 3 pair doesn’t hit on the flop, there is a good chance it will be folded while the suited connector often will flop a straight or flush draw along with possibly hitting a mid-rank pair, a total probability higher than the 12% chance of the 3-pair setting.

This is not to suggest that Equilab is not useful. Showdown equity provides a first cut to your chances of profiting. And, by recognizing that folds, either by you or your opponent is possible, it’s value can be used as the baseline for equity adjustment depending on the particular circumstances. It would be nice if an equity calculator can also provide realized equity but that would most like have to rely of analyzing many millions of hands under many varying conditions for there is no theoretical or simulation way of obtaining fold equity as far as I know.

As a final point, estimating your equity should not only take into account future folding possibility, by both you and your opponents, but also their calling range as the hand proceeds. An opponent may limp preflop with a top 80% range, but depending on the board, his calling range will almost surely narrow unless he is a maniac bettor. How much it narrows is a challenge a poker player must address.

In the previous post we examined how suit interactions can influence showdown equity with pair vs. pair. We found that there is not a great effect with the King vs 44 showdown equity ranging from 82.58% to 80.7% for the kings. In this post we again look at suit interactions but for AK vs 22, which serves as an example of an unpaired Broadway hand versus a low pair. Without any suit specification, 22 has the advantage 52% to 48%. However, AKs vs 22, is just about a 50-50 case.

Now, let’s look at specific suit interactions with the AK equity stated first. We also include the above results:

Any AK vs any 22: 48.0 - 52.0

Unsuited AK vs any 22 47.4 - 52.6

Unsuited AK vs no common suit AdKh vs 2c2s 47.0 – 53.0  (worst for AK)

Unsuited AK vs one common suit: AdKh vs 2d2s 47.4 -52.6

Suited AK vs any 22: 49.9 -50.1

Suited AK vs one common suit: AdKd vs 2d2c  49.7 - 50.3

Suited AK vs no common suit: AdKd vs 2c2s: 50.8 vs 49.9 (best for AK)

As we saw in our previous post, there is not a large difference in equity for the various suit interactions. In all cases but one, the low pair had slightly better equity, being as high as 53% for AK unsuited and 22 with different suits. This is to be expected because the only way 22 wins if AK pairs or both hit sets is if it flushes (ignoring straights). Also, as expected, AK does best when suited with 22 not having the AK suit, again flushes are important.

Introduction. It is well known that if two players each have a pair, then the player with the higher pair is approximately a 4 to 1 favorite if the hand went to showdown (80% equity vs 20% equity).  The reason it is approximate is because there are suit and rank interactions affecting flush and straight possibilities.

Analysis, Using the Equilab poker calculator I analyzed the suit interaction effect for KK vs 44. Here there are only two possibilities for a straight with the kings, but 4 for the fours. If you enter KK vs 44 in the calculator, you will find that the kings have an 80.93% chance of winning, which, of course, is due to its higher rank if both set or do better or if neither set. For this comparison, only one of the two hands can straighten. If the comparison was KK vs QQ, the kings have a slightly better winning chance of 81.93% because if the board provides a straight for both, the kings would either take it or chop.

Results. Now for the suit interaction evaluation. We need only consider one case for the king’s hand, say Kc Ks.  We can then compare that to all canonical possibilities for the fours, two suits the same, one suit the same, no suit the same. The possibilities and resulting showdown equities for the kings are given below:

1) Both suits the same (4c 4s) eq=82.58%

2) One suit the same (e.g., 4c 4d) eq= 81.92%

3) Two different suits (4d 4h) eq = 80.27%

It should be apparent that the kings have the highest equity for 1) Equal Suits since the kings always win if the board provides a flush. In a similar fashion, the fours do best when they have two suits different than the kings for that allows them a possible flush win. The equity gain for the fours with two different suits compared to two equal suits is 2.31%.

Conclusions. What have we learned from this? Not much IMO mainly because it is consistent with what one would suppose and, in practice, guessing opponent suits is not a common or likely successful action in most cases. But, since this blog site focuses on poker math, it may be of interest to nerds like me.

One other thought. While the 80/20 is a reasonable approximation for all suit interactions, in reality it probably is conservative if one considers realized equity. Someone holding a king pair is unlikely to fold a flop while a four-pair holder might very well do so unless he sets or has a good possibility for a flush or straight. If so, then the four pair has 0 equity.

In keeping with my past attempts to simplify EV analysis, I offer the following:

If you are pretty confident about a single opponent’s opening range, then one strategy option is to call with the same range.  If you do so, your equity is 50%, so the EV equation in pot size units is as follows:

EVcall = 0.5*(1 + Bet) - 0.5*Bet = 0.5.

This equation used the setting of the strategy as the decision point, so hero’s investment is the bet plus the blind amount. An EV of 0.5 (half the pot) is to be expected if both players have 50% equity. Note that the EV is independent of the bet size and, if not a river decision, ignores future betting or assumes the hand is checked down.

If you raise rather than call, say to K*Bet, K>1, and villain fold equity is fe, then

EVraise = fe*1  + (1-fe)*(0.5*(1+ K*Bet)- 0.5*(K*Bet)) = 0.5*(1+fe)

If fe > 0, a raise results in a higher expected value than calling. If villain always folds (fe=1), EV = 1; if villain folds half the time, EV = 0.75, and, as shown by the previous equation, if villain never folds (fe=0), EV = 0.5. Again, these results as shown do not depend on the bet or raise size, but generally fe will be an increasing function of K and may dip for large K if villain suspects a bluff.

Note that since EV is not dependent on the raise size, it is suggested that you probably should raise big to increase the fe value, especially if your hand is a good one.

Of course, if your hand is at the higher end of the bet/call range, then the above results are conservative and they are optimistic if your hand is at the lower end of the range.

Counter Argument: If your equity is 50% when your calling range is equal to villain’s betting range, why not call with a tighter range so your equity is greater than 50%.

Answer: Following this reasoning, why not call only when you have aces for that will maximize your EV. True, but it only maximizes it for the hands you play and you will hardly play at all, so your EV is quite negative since you’ve been paying the blinds and possibly antes all the while. Of course, if the pot is large enough, pot odds theory says you can profitably call with realized equity less than 50%.

Summary. Have we found a simple solution to always be a winning player? Of course not. For one, you will never know your opponent’s range. And even if you can estimate how often he plays, say 20% of the time, his 20% hand range may (probably) not coincide with your 20% hand range. Even if you can accurately estimate opponent’s range, that doesn’t mean you should always call or raise his bet with that same range. If you have one of the lower ranked hands in the range, a fold may be best.  But, as with so many math-based strategies, this type of analysis can offer initial guidance to better playing, which can then be modified to include other factors not considered here.

*********************

Addendum: We’ve shown calling at villain’s betting range has EV = 0. 5.  We can also call at a different range if we want to set EV = E, say. Let E(Rh,Rv) be the equity of calling with hero range of Rh when villain is betting with range Rv. Then

EV = E(Rh, Rv)*(Pot + 2Bet) - Bet = E

E(Rh,Rv) = (E + Bet)/(Pot + 2Bet)

For the ranking system I used, the following table shows the calling range to use for various bet sizes that results in a showdown EV equal to 0, thus assuring no loss. These results will vary somewhat for other ranking systems but should serve for general use.

Example: If villain makes a 2xPot size bet with a betting range of top 10%, to assure +EV (E=0), hero needs equity of 40% and achieves this with a top 26% range. If villain bets less than ¾ Pot with this range, hero can always call. Naturally if hero’s hand is at the bottom of a range, discretion should be used.

We note that in an earlier post (HM Quickie 15) we did a similar analysis but there we required that every hand in hero’s range had the required equity.

It should be apparent that the larger the number of players pre-flop, the more likely it is that one or more has a good hand. For this reason, you often see advice/admonitions such as “don’t bluff against more than 1 or 2 players,” and “It’s not how many that folded before you that is important but how many are behind you” and “it is usually advantageous to play most every hand in a heads-up game.”

What I have not seen is any type quantification of how opponent-hand goodness relates to the number of players. So, after quickly deciding I could not do much mathematically, with much difficulty I wrote a simulation program to provide quantification insight.

The first question I considered, is the definition of a good hand. Clearly this is subjective; a pair of threes, heads up, is pretty strong with showdown equity of 53% against your random opponent, but not so against a full table (equity = 13%). Also, a pair of threes could be preferred against 65 suited but that is very doubtful against AK suited. Nevertheless, to keep things fairly simple, I used the following to define a good hand, listed in strength order:

-Pair, -Two Picture Cards (A-J), -Suited Connector, -Suited, -Connector

I then simulated dealing hands against 1 to 8 opponents. The table values show the probability that at least one of the players has that hand type and no one has higher. No consideration is made for your holding.  If Player 3 has a connector, but Player 5 has a pair, I tabulated that as at least a pair occurring. In other words, the values refer to the best hand type as I defined “best.” Clearly, more than one player in a hand can have one of these hand types. The last row shows the probability that none of the players hit

Example: If you are in a 6-max Hold ‘em game UTG, then after the deal, the probability that at least one of your 5 opponents has a pair is 26.1%, or the probability is 24.6% that at least one player has a suited hand but no one has better. The probability that none of your 5 opponents have any of the indicated hand types is only 5.6%.

Note: As the number of players increase, probabilities go down for all hand types but pairs; this is so because if a pair is dealt, no account is made for occurrence of weaker hands. Of course, conversely, since a pair is considered to be the best hand type, the pair probability increases with N. Also, as expected, the chance that at least one player has a good hand increases with N, so for 8 opponents, it’s about 99%. Finally, no account is made for card removal if you are considering only players behind. If several players have entered the pot before you, that has a small effect on cards players behind you might have, the so-called bunching effect.

Application: I’m not sure there is a direct application except for very special situations and even then, it would have to be an approximation since no account was made for what you hold along with many other factors (e.g., opponent tendencies). However, I think it does provide useful insight. For example, the probability that your one opponent has a pair is less than 6%, so if you hold two picture cards or a suited connector, you are in good shape. However, if in a full ring game, the probability at least one of 8 opponents has a pair increases more than 6-fold to 38.3%, so your pair of 3’s is not so hot and you see many players in early position folding 3’s in full-ring games. Also, it is apparent that with 4 or more opponents who have not yet acted, it is very likely (>90% chance) that at least one has a hand that might be troublesome.

The above conclusions are not that surprising but quantifying relationships can provide useful information for making good decisions, such as developing an initial cut at estimating EV.

What if the stakes you play at are inconsequential? Winning (losing) a $10 pot with blinds of 1 2 gives you the same pleasure (pain) as you get winning (losing) a $100 pot with blinds of 10 20. In other words, the win/loss - pleasure/pain relationship depends primarily on your good or bad decision making and not on the win/loss amount. If that is the case, how does that affect how you should play?

For example, if someone plays at a stake that is much higher than his bankroll dictates, he might play poorly because losing will be financially painful. On the other hand, if the stake is such that there is no financial stress to the player with a loss, then his play can be much more aggressive and that may lead to more profit if he uses the aggression correctly.

It seems pretty obvious to me that playing at a stake higher than you can afford will likely be a bad choice because it can negatively affect your play. For example, you might be behind with card equity in the 40% range having to call a big bet. Yet the pot is big enough so that a call is the right EV decision, but mathematically you wrongfully fold since losing a big bet will cause a significant dent in your bankroll. Playing scared is never a good strategy. On the other hand, playing at a stake much below your bankroll level is also a poor choice if winning money is your primary objective.

The bottom line is to seek a stake level that maximizes your playing objective – money or winning or perhaps some combination of the two. A previous post, Bankroll Management and Risk of Ruin (RoR), provides further information on this issue for cash games and 9 man Sit’n’Go’s.

It is a general rule that you should not try to bluff more than 1 or 2 opponents. Therefore, you are more likely to find bluffing opportunities in a Heads-up game than in a 6max or full Ring game. But, for a Ring game, what if all opponents but one fold? Then you are essentially heads-up so should that be equivalent to a two-man game? The answer is no and here’s why.

In a Heads-up game, neither player is likely to have a strong hand so players are forced to play weak hands more often. Therefore, with a weak hand, they will often have to win by inducing a fold through bluffing. In a game with more players, it is likely that at least one has a strong hand so a bluff is less likely to succeed. Thus, in a Heads-up game, your one opponent probably has a weak hand whereas in a Ring game, your one opponent probably has a strong hand. The former calls for bluffs while the latter suggests otherwise unless special conditions exist.

For a final thought, assuming a bluff has zero equity, the general rule for bluffing is the following:

If villain fold frequency is greater than Risk/(Risk + Reward), then a bluff is profitable, where Risk is the amount you bet and  Reward is the pot before you bet; thus –

Bluff if Fold Equity > Bet/(Pot+Bet).

The following table/graph displays some results.

Example: If you bet 0.75 Pot, with an air-bluff hand, you need fold equity of at least 43% to be profitable.

A poster asked if you should value bet hands that beat at least half of those in villain’s range but your equity is less than 50%.

The poster’s question might involve a definition issue. In evaluating your equity against a villain’s range, you have to account for occurrence frequency of hands in the range. How many of those range hands you beat is not meaningful if the frequency of their occurrence is not accounted for

It’s not clear if the poster considers frequency. For example, 77 can be said it have 50% equity versus a range of {88, 66}. But, that is so only if 88 and 66 are equally likely to be in the range. If villain always plays 88 but may not always play 66, then the 50% equity estimate is incorrect

Typically a value bet is defined as one where you think you have the best equity; that means it has to be greater than 50% assuming one opponent. In such a case, you want to bet as much as you will think opponent will call for that will maximize EV. For example, suppose you think your equity is about 75%. You guess that if you make a bet of 1.5 Pot, villain will most likely fold, so you win the pot. But, if you make a pot size bet, villain will call 50% of the time. With that call probability, your EV is 1.13 Pot[1], 13% better -- a significant advantage.

As a simple example of an EV-based call decision, if the pot odds on an all-in bet are 3 to 1, the EV math says you can profitably call if you estimate your equity to be at least 25%.

Similarly, if you have the lead, you can use EV math to find the maximum bet size when you have equity less than 50% assuming villain always calls. The math says you should then bet no more than   Pot*eq/(1-2*eq). For 25% equity, bet no more than half the pot.

As with all math-based analyses, adjustments should be made to account for factors not explicitly considered. For a more complete analysis, stack and pot sizes along with fold equity should be considered in an EV context for good decision-making. Here, we assumed villain always calls, but with equity less than 50%, it might be better to bet big if it is likely villain will fold to such a bet.

If poker were easy, everybody would win.

-------------------------------------------------------------------------

Click the Archive button at screen top to access other posts.

Introduction

In the process of developing a fold equity software program, I looked at extreme values for card equity and fold equity to see how they affect EV. The following formula is applicable for a hero bet assuming villain’s call amount equals the bet amount:

EV = FE * Pot + (1-FE)*(eq*(Pot + 2*Bet) – Bet)

where

FE = fold equity

eq = card equity

Pot is the pot amount before hero bet

Bet is the bet amount which villain must call

FE for Equity Extremes

The following shows EV outcomes for the extreme values of FE and equity and for each at 50%, assuming Call = Bet.

  We note the following:

Hero EV is always negative if both FE and eq equal 0%, since villain never folds and hero never wins

Hero EV always equals the pot if villain always folds

If hero equity = 0 he can still have positive EV if villain folds enough. For FE = 50%, hero EV is positive if the bet is less than the pot

EV is always positive for equity >= 50%. At equity = 50%, EV cannot exceed the pot and at equity = 100% EV will always be at least equal to Pot.

If villain never folds, FE=0, hero’s EV range is -Bet to Pot+Bet as his card equity ranges from 0 to 100%

Minumum FE for +EV (EV>=0)  When Card Equity < 50%

It can be shown that the critical minimum fold equity value to assure +EV when your card equity is less than 50% is the following:

FEcrit = (Bet-eq*K)/(Pot+Bet-eq*K),

where K = Pot + Bet + Call

Example: If hero bets 30 into a pot of 40 and he estimates his equity to be 25%, then the minimum fold equity he needs assuming villain must call 30 is as follows:

K = 40 + 30 + 30 = 100

FEcrit >= (30 – 0.25*100)/(40 +30- 0.25*100) = 5/45 = 11%

So, even though hero has only a 25% chance to win at showdown, if villain folds at least 11% of the time to a 3/4 Pot bet, hero will show a profit.

The table below present the critical minimum fold equity for various card equities and bet sizes. Blanks mean EV >=0 for any villain fold probability. Note that for card equity > 50%, hero always has positive EV. Even if hero has complete air, he can still have positive EV if villain folds enough, the underlying case for bluffing along with the need to have balance.

It is seen that for the low card equity situation, the minimum fold equity increases with the bet size. That is somewhat tempered by the fact that a player is more likely to fold as the bet size increases. Here is where the art of bet sizing comes into play.

Summary

Folding is an important factor is doing an EV analysis. We showed in this article how it interacts with card equity and bet size to define EV. Fold equity is included in a number of other blog posts. Four in particular are the following:

When Is A Villain Fold Better Than A Call?     

Bet Sizing. Two Case Re Fold Equity.

 EV Analysis When Fold Equity Is Greater Than 50%

The Cost of Folding

Introduction. A previous post described 6 methods for estimating equity, a factor involved in most EV analyses. In this post we applied the use of the Equilab equity calculator to provide results several types of hands. They are as follows:

Pairs: AA, JJ, 88, 44     Premium Non Pair: AKo   Suited Connectors: QJs, 87s   Connector: JTo   Blah: 95o

Hand Showdown Equity. The showdown equity of these hands were evaluated against one or more villain ranges as follows, where the percentages are the fraction of total hands ranked by effectiveness according to Equilab:

Nit: 5%, 10%   Moderate: 15%, 20%, 25%, 33%   Loose: 50%   Maniac: 75%, 100%

The results in tabular and graphic form are as follows:

Example: QJs against a moderate range player has showdown equity of 44%. AA has about 85% against any range while rag hands like 95o has an equity range of approximately 25% to 40%, Nit to Maniac.

Equity Realization. A related and important concept is equity realization, the chance that the showdown equity will be realized. Of course, this is true if the hand gets to showdown. Consider the hands 44 and QJs. Each has showdown equity of approximately equal win probability for each villain type. Now, if the flop doesn’t hit the small pair with a set, the player will likely fold to any aggression. However, the suited connector player may elect to continue for there might be a reasonable chance of flushing, straightening or winning with a premium pair. The suited connector has more “playability.” In essence, the small pair player is more likely to fold than the suited connector player and therefore the pair doesn’t realize its equity. To provide some backup to this idea, Equilab rates the QJs as a top7.4% hand while 44 is rated as top 36.4%.

Discussion. The above graph presents no surprises, in my opinion, except possibly for AA having about the same equity for all villain types. In all cases, equity increases, however slightly, as the villain range increases. Playing against maniacs will likely do better than against nits but care is required and you have fold equity against nits. For hands in approximately the same rank range, the pairs do better than the connectors. Although not shown here a suited hand will do about 3% better than a comparable non-suited hand.

An implied odds strategy is one where you might make a -EV call on the current street because if you hit your draw on the next street you can profit enough to justify the call by making a large enough bet that wins often enough.

I discussed in a series of past posts an advanced implied odds model that includes hero hitting his draw but not always winning (reverse implied odds). (Link is at end of article). The model also includes the possibility that villain will not always call your future bet.  A typical application is to determine the future bet required and associated implied odds to assure a profit. In this Quickie, I examine how the future bet requirement is influenced by villain call probability.

I looked at draws for a set, OES, and flush. For the set and straight, I assumed a win given hit probability of 75% and a value of 85% for the flush. I also assumed villain makes a pot size bet on the current street so required showdown equity for a hero call is 33%, which is more than the assumed hands have.

I found the following:

-          The future bet (FB) requirement increases as villain call probability decreases, even though a villain fold assures a win while a call does not. FB increases dramatically for very low call probabilities.

 -          If villain will never call, you cannot justify the current street call, because you will not hit often enough with these draws

 -          Regression fitted equations for each case showed excellent fits to the model results (R^2 > 99%)

 Set:     FB=12/Call%

OESD: FB = 6.6/Call%

Flush: FB = 3.5/Call%

The following shows these results graphically:

As an example of possible application, suppose you have a flush draw on the turn and the values I used apply. Based on past experience and/or HUD data, you estimate villain will call your river bet if you hit only 30% of the time since the board will certainly hint of a flush. In that case, you need a future bet of 3.5/0.30 = 11.7 Pot to justify calling the turn pot bet. If the effective stack is less than that, a call is not justified on an expected value basis.

Naturally these equations apply only for the values chosen such as the villain bet size and win given hit probabilities. I do think this kind of exercise is useful for much has what has been written about implied odds is often based on gut feel rather than math analysis.

Cliffs: If villain makes a pot size bet, the future bet (FB) needed is as follows:

Set: Assumed Win% (Hit * Win|Hit) = 0.12*0.75 = 9%: FB= 12/Call%

OESD: Assumed Win% = 13%: FB= 6.6/Call%

Flush: Assumed Win% = 16.6%: FB = 3.5/Call%

One of the ‘good’ things about poker math is that almost always there are several ways to calculate a probability. However, there is a negative to that, namely, if one is used to method A and someone in a discussion uses method B, then that may be confusing.

In this article we present various methods for determining poker probabilities. We will answer the following question to show the methods:

What is the probability that you will be dealt at least one ace in a hold ’em hand?

Some methods will be fairly obvious and perhaps some may be puzzling. At the end, we will pose some problems and suggest you try to work out the answers several ways to help in understanding each method. Answers are provided at the end of the article.

As a reminder, for card deals, we define the probability of an event as the number of ways or outcomes the event can occur divided by the total number of outcomes. For a joint event, say the simultaneous occurrence of 2 events, A and B, we have Pr(A & B) = Pr(A) *Pr(B|A), where Pr(B|A) is the conditional probability of B given that A has occurred. If A and B are independent—the occurrence of one has no effect on the occurrence probability of the other-- then Pr(B|A) = Pr(B). As a result, for independent events, Pr(A & B) = Pr(A)*Pr(B).

For either of two events, A or B or both, Pr( A or B) = Pr(A) + Pr(B) – Pr(A & B ). If the events are mutually exclusive (cannot occur simultaneously), then Pr(A & B) = 0, so Pr(A or B) = Pr(A) + Pr(B).

Notation for dealing the hold ’em hand and looking for aces:

A1 = Ace on first card; A2=Ace on second card

X1 = any non-ace on first card; X2 = any non-ace on second card

Joint Probability Method: Ace on only the first card or Ace only on second card or Ace on both cards

Pr(Dealt an ace) = Pr(A1 & X2) + Pr(X1 & A2) + Pr(A1 & A2).  (Note that the three joint events are mutually exclusive)

The deck has 4 aces and 48 non-aces. Therefore,

Pr(A1 & X2) = 4/52 * 48/51 = 0.0724  (48/51 is a conditional probability)

Pr(X1 & A2) = 48/52 * 4/51 = 0.0724

Pr(A1 & A2) = 4/52 * 3/51 =  0.0045

Pr(Dealt an ace)= 2*0.0724 + 0.0045 = 0.1493.

Complementary Method: One minus the probability no ace is dealt

Pr(Dealt an ace) = 1 - Pr(X1 & X2)

Pr(X1 & X2) = 48/52 * 47/51 = 0.8507

Pr(Dealt an ace) = 1- 0.8507 = 0.1493

Conditional Probability Method: Ace on first card or no ace on first card and ace on second card

Pr(Dealt an ace) = Pr(A1) + Pr(X1)*Pr(A2|X1)

= 4/52 + (48/52)*(4/51) = 0.1493

Inclusion-Exclusion Method: P(A1 or A2) = Pr(A1) + Pr(A2) - Pr(A1 & A2)

Pr(Dealt an ace) = 1/13 + 1/13 –(1/13)*(3/51) = 0.1493

Combinatorial Method: Pr = Number of combos with at least one ace/Total number of combos

C(n,r) = n!/[r!(n-r)!], where m! = m * (m-1) * (m-2) *  . . . * 3 *2 * 1

Number of Single Ace Combos = C(4,1)*C(48,1) = 4*48 = 192 (1 of 4 aces and 1 of 48 non-aces)

Number of AA Combos = C(4,2) = 6 (2 of 4 aces)

Number of Hand Combos = C(52,2) = 1326 (2 of 52 cards in the deck)

Pr(Dealt an ace) = (192 + 6)/1326 = 0.1493

Permutation Method: Number of perms (ordered events) with at least an ace / Total number of permutations

P(n,r) = n!/(n-r)!

Number of Single Ace (AX) Perms =2* P(4,1)*P(48,1) = 2*4*48 = 384 (the 2 is for two positions the ace can occupy)

Number of AA Perms = P(4,2) = 12

Number of Hand Perms = P(52,2) = 52*51 = 2652

Pr(Dealt an ace) = (384 + 12)/2652 = 0.1493

Summarizing -- we used 6 different methods to calculate the probability you will be dealt an ace. It is advisable to use at least two methods in doing your own calculations for probability is a tricky animal and therefore a check can often reveal an error.

Problems

1.       What is the probability you will be dealt a pair of aces?  

2.       What is the probability you will be dealt exactly one ace?  

3.        What is the probability villain has a pair of aces

a.       given you don’t have an ace?  

b.       given you have one ace?  

4.       In a heads-up game, what is the probability neither player has an ace?

5.       What is the probability the flop has exactly two aces?

6.       What is the probability the flop has two cards the same suit?

7.       Given you have 88, what is the probability you will flop a set or better (ignore straights and flushes)?

8.       Given you have 88 and no overcard on the flop, what is the probability there will be an overcard by the river?

9.       Given you hold 2 spades and the flop has 2 spades, what is the probability you hit a flush by the river.

10.   What is the probability villain has a pair

a.       given no other specification

b.       given you do not have a pair?

c.       given you have a pair?

Bonus Question. 8 is the median rank. In a heads-up match, you are dealt 8 X, X< 8. What is the probability your 8 outranks your opponent?

Answers (using one of the above methods; can be done other ways):

1.  4/52 * 3/51 =0.45%

2.  4/52 * 48/51 + 48/52 * 4/51 = 14.48%

3.   a. 4/50 * 3/49 = 0.49%

     b. 3/50 * 2/49 = 0.24%

4.  C(48,4)/C(52,4) = 71.9%

5.  C(4,2) * C(48,1)/C(52,3) = 1.30%

6.  4 * C(13,2) * C(39,1)/C(52,3)= 55.1%

7.  1 - C(48,3)/C(50,3) = 11.8%

8.  24/47 + 23/47 * 24/46 =76.6%

9.  (9*38 + 38*9 + 9*8)/(47*46) = 0.35

10. a. 13 * C(4,2)/C(52,2) =0.0588

      b. [11 * C(4,2) + 2 * C(3,2)]/C(50,2) = 0.0588

      c.  [12 * C(4,2) + 1 * C(2,2)]/C(50,2) = 0.0596

Bonus: Prob = Chance of 2 cards less than 8 = C(23,2)/C(50,2) =  11*23/1225 = 253/1225 =0.207

Introduction

“What is a good bet?” is a question discussed in many introductory poker discussions. The answer usually is something like this:

A good bet gets better hands to fold and worse hands to call.

A sometimes suggested third answer is to bet for protection, a bet made with a vulnerable hand to get a drawing hand to fold or to call with incorrect odds. Also, information bets and blocking bets have been proposed for certain situations.  In this discussion, we will concentrate on betting to get worse hands to call so that your EV exceeds that with a fold.

A Worse Hand and Realized Equity

The first thing we’ll consider is to define a “worse” hand. An obvious definition is to relate the “worse” aspect to equity, the chance of winning the hand. Say you have 22 and you know villain has AKs. By a fraction of a percent, 22 is favored, so by a small margin, AKs is worse. Would you proceed to play 22 as a potential winner based on that? I doubt it and here’s why - realized equity. If you don’t hit the flop with your very small pair, and there’s only a 12% chance you will hit a set or better, what do you do if an ace or king falls and your tight villain bets aggressively? Fold is probably the right decision assuming other factors don’t dictate otherwise.  In fact, you will likely fold to most any missed flop containing one or more premium cards and villain betting. A fold of course means you will not realize your equity. Having AK,s however, you may hit a high pair and the hand has good possibilities for improving to a flush or straight. Even the Ace could win. That is why AKs is rated much higher than 22 even though the latter has a slightly better showdown equity preflop.

For any analysis which includes a showdown equity estimate, that estimate should be adjusted to the extent possible to reflect realized equity. You will increase equity if incorrect villain folds (folding hand he would have won) exceed your incorrect folds; your equity will decrease if the opposite is true.

The Minimum Bet Size Equation

Let’s now suppose you have a much better hand, say you are a 60/40 favorite after the flop or turn, a very good situation and there is a good chance that the equity (or better) is realized. You have the lead. You can check, bet small or bet big. A check would be a consideration if it induces villain to bet. However, it also opens the possibility for a check-back which could give villain a free chance to hit his good draw. Against very aggressive villains this might be a good action, otherwise not so good. Betting small has a similar consequence but at least builds the pot some with a winning hand. Betting big has the desirable result of a much bigger pot but might result in a fold that gives you a minimum win amount. Is there a minimum bet size that provides a potential bigger pot but gives villain the wrong pot odds to call?  Yes, it is given by the following equation:

Bet size for Villain Incorrectly Calling > Pot*(1-eq)/(2*eq-1),  eq > 50%.

Example

For our example case of hero being a 60/40 favorite, assume the pot is 20. Then for villain to incorrectly call, a bet has to be at least the following:

Bet size > 20*(1-0.60)/(2*0.60 – 1) = 8/0.20 = 40.

Any bet greater than 40 will more than triple the win amount, but it will also insure that villain has incorrectly called when he is the underdog. This equation was derived so that on average hero wins at least as much as he would if villain folded. To see this, here is the EV fold and EV bet equations, the latter based on the assumption that villain folds on the next street if the current street is not the river:

EV if villain folds on current street:  EV= Pot = 20

EV if villain calls a bet of 40 but folds next street: EV = 0.60*(Pot + 2*Bet) - Bet = 0.60*100 - 40 = 20

The equality of the two EV’s is as expected for it shows that any hero bet greater than 40 that is called will do better EV-wise than a villain fold. Of course, making a very large bet increases the chance of villain folding, so here is where a read of villain’s tendencies and factors such as position and stack size are very important to guide you to making that bet which does not give villain good pot odds but keeps him in the pot.

Table and Graph

The following table and graph show minimum bet results in pot size units for various equities over 50%. Multiply the given value by the pot size to convert to actual units

Example: If you estimate your realized equity to be 70% on the river, to have villain incorrectly call you need a bet of at least 0.75 *Pot. Thus, if the pot was 100, your bet should be at least 75 or else you would rather have a fold.

Check: Villain Fold: EV = 100. Villain Call: EV = 0.70*(100+150)-75 = 175-75=100

Summary

Here’s what we can conclude from the above:

1.       Raw, hot or cold and showdown equity are terms to describe your winning chance if the hand went to showdown. An equity estimate based on showdown should be adjusted to what can be described as realized equity to reflect the potential you or your opponent might incorrectly fold on following streets even if ahead.

 2.       Being ahead you generally want your opponent to not fold if that will result in a larger winning pot in EV terms. We provided the minimum bet size to achieve the objective.  Of course, betting too large will likely result in a fold, so the non-math art of poker becomes an important factor.

 3.       If the bet is not a closing one, results should be adjusted for factors not explicitly considered.

The Independent Chip Model, ICM, is used to analyze tournament dollar performance taking into account the prize structure. The basic approach is to convert one’s stack size in chips to his winning chances for the various money placings (1st, 2nd, etc.). This in turn determines the winning amount. The simplest case is that at any point in time, the chance of winning 1st place is equal to your stack size divided by the total of all chips in the tournament. There are ICM calculators on the Internet where you can input stack sizes for a small number of players and the profit structure to determine the expected return for each player.

The underlying assumption of ICM theory is that players are all of equal capability. That, of course, is almost certainly not true and we discuss here one way to account for varying capability.

So, how can we adjust the model to account for someone of better or worse capability and still retain the ICM approach? We claim that assigning capability factors, CF, to players is one such way. Say, 1.0 is the CF for an average player. If a player is ‘twice as good’ as average, his CF is 2.0. A poor player may have a CF of 0.80. Then instead of using stack sizes as the ICM profit-determining factor, use CF*Stack Size, an adjusted stack size that accounts for varying capability.  A natural metric for assigning CF’s would be relative Return on Investment, ROI, where ROI is total tournament profit or loss divided by total investment.

However, negative ROI values are a problem.  A comparable metric that avoids this is a player’s expected return relative to that for a player with CF=1. This can be gotten from an ROI measurement as follows:

If a tournament prize amount per player is P and the fee is F (Total Buy-in = P+F), the expected return with equal player capability is P (each player can be expected to lose the fee, F). If a player over a number of such tournaments has ROI= R, then his average return is (1+R)*(P+F). Therefore, the capability factor is

CF= (1+R)*(P+F)/P  = (1+R)*(1+F/P).

Note the ROI for the average player is -F/(P+F) for which his CF= P/(P+F)*((P+F)/P) = 1

Example 1: Over ten $9 + $1 tourneys (P=9,F=1), a player has an average profit of $22 with a total investment of $100. His ROI=22/100 = 22%. Therefore, CF =(1+0.22)*(1+1/9) = 1.36.

Example 2: Over a ten 9+1 tourneys (P=9,F=1), a player has an average profit of -$22 with a total investment of $100. His ROI=-22/100 = -22%. Therefore, CF = (1-0.22)(1+1/9) = 0.87.

Example 3. If both of the above players were in the same 9 player tournament with the other 7 players having CF=1, the modified ICM model would give an expected profit of $1.55 to the best player, -$1.19 to each of the 7 average players and -$2.23 to the worst player assuming a 50-30-20 prize structure. Total profit/loss = -9, the total fee for 9 players.

Of course, one can use other data such as winning amount history available on some sites or use a subjective CF representing the edge you think  you have over the opposition.

Assessing how useful and practical this model is I’ll leave to others.