KKing David

Ruminations on poker

Archive for the tag “poker math”

Math, Combinatorics and Frequencies

NOTE:  This entry was originally posted on a different site on June 16, 2016 and has been slightly edited prior to re-posting here.

I played this hand at a private cash game a few days ago.  On the river, it was obvious that I needed to fold.  Then again, maybe not.

With QQ in middle position, I raised to 8 BBs following a single limper.  This is a bit more than normal for me, however at this game there were frequently multiple callers pre-flop so I decided to let them pay me a little extra.  Or thin the field.  Either would be fine.

There were 3 callers, making the pot 34 BBs, already a bit bloated.

On a flop of K54 with two spades, I bet 16 BBs and the button called.  For purposes of this blog, I’ll call him “Adam.”  Adam is a 23-year-old loose/aggressive thinking player.  He likes to play mixed games and finds Texas Hold’em a bit boring, but plays it because that’s what’s available around here.  Although I’ve only played with Adam a few times, he is willing to mix it up, gamble and be aggressive if he senses an opportunity to steal a pot.

One the one hand, I’m targeting a hand that will call my slightly less-than-half-pot bet like 66-JJ, A5s, A4s or 56s.  On the other hand, I’m concerned about Adam holding either a King or a flush draw with two spades in his hand.  If he does have a King, it’s probably not AK as he would be more likely to re-raise pre-flop on the button.  But it could be KQ or KJ, maybe as weak as KTs.

The turn is an off-suit deuce.  I bet 20BB more and he calls.  My bet is deliberately small, hoping he will call with weaker pocket pairs or other non-flush draw / non-Kx hands that might fold to a larger bet.  He will definitely call with a flush draw and that’s a risk I’m willing to take.  If he has a King with a strong enough kicker, he might raise just in case I am the one chasing a flush.  I know he’s capable of playing his draws aggressively, but not 100% of the time.

The river is the 8 of spades, which completes the flush (if that’s what he’s chasing).  Now I’m faced with a situation that Bart Hanson at Crush Live Poker calls “5th Street Chicken.”  This is where I’m out-of-position, and don’t want to put any more money in the pot.  But if I check, I’m opening the door for Adam to bluff if he actually has one of the hands I’m targeting.

Quick recap:  The board is Ks 5h 4s – 2d – 8s.  I have QQ.  Adam is on the button and called my pre-flop raise, and called my flop and turn bets.  There is now 106 BBs in the pot.

The pot is really too big for my 1-pair hand.  I don’t even have top pair.  If I bet on this river, am I essentially turning a hand with showdown value into a bluff?  Yes.  Is that a good idea?  No.

I check.  Adam bets 32 BBs.

Clearly I have to fold.  He either has the King, or he hit his flush draw, and he’s betting for value.  His bet is small, which it has to be after I waved a white flag by checking the river, all but announcing that I don’t have a flush, nor a hand that is strong enough to bet/fold (for value).

What would you do?  Leave a comment below…


I decided to fold, but before relinquishing my cards started doing the math.  There are 106 BBs in the pot.  Adding his bet of 32 BBs makes it 138 BBs.  By calling, I’d be risking 32 BBs to win 138 BBs means I’m getting pot odds of 4.3-to-1.  I would have to win 1 out of 5.3 times for calling to be profitable, in the purest poker mathematics sense.  That’s slightly less than 19%, a pretty low threshold.  I was recently reading a limit Hold’em strategy book, and recalled some commentary about calling on the river.  Often you will be getting pot odds of 10-to-1 or more in a limit game due to the constraints on bet sizing.  The author’s point was that while most players should fold much more often pre-flop and on the flop, they should call on the river when they have showdown value and there is any chance they are good as little as 8-10% of the time.  That’s just how the math works.

So I ponder this for another minute.  Adam is capable of turning a weaker hand into a bluff here.  What does he think I have?  I went bet-bet-bet-check.  My range can easily be 99-QQ, AK, KQ, KJ, and is probably pretty transparent at this point that I have a 1-pair type of hand.  After I checked the river, the 3rd spade coming in is a great bluffing card for him.  With a King, he’s more likely to check back after the scare card arrives.  But I think he’s figured out that I can fold when it’s obvious that I’m beat.  And he’s got the stones to take advantage of my discipline and tight image.


I don’t do the combinatorics at the table, but there are far fewer flush draw combos in his range than other combos.  If I include literally any two spades that include an Ace or have two gaps or fewer, that is about 24 combinations.  Plus 3 combinations of pocket 88’s that binked the river for a total of 27 value combos.  I’ll assume that he always bets with these hands.

His non-flush draw, non-Kx calling range for the flop & turn (66-JJ, A5s, A4s, 65s)  has about 39 combinations.  These are the hands that I beat and was targeting with my small bet-sizing.


Now the questions is how frequently will Adam turn one of these 39 hands that I beat into a bluff on the river?  If the answer is 7 or more out of 39, then the mathematically theoretically correct response for me is to call.  Add 7 bluffs to the 27 value combos, and I win 7-out-of-34 times, or 20.6%.  If I do call his river bet, I’m probably going to lose this pot, but poker theory tells me I’ll lose less over the long run by calling in this spot than by folding.  It feels really messed up to have to think this way.  Nevertheless, I think his bluffing frequency with this set of facts is greater than 7-out-of-39, probably closer to 13-out-of-39 (33.3%) or more.  I feel like I’m about to throw away 32 more BBs because of poker math.

To succeed at no limit Texas Hold’em, however, you have to trust your reads.  My read on Adam says his bluffing frequency is high enough for me to call.  And my read on this situation overall is that the most important question right now is the one about his bluffing frequency.

So I call.

He says “I have a pair of fives.”  With 6-5, the turn card gave him a gutshot straight draw to go along with his weak pair… just enough for my small bet to keep him in.

When he sees my cards, he says he figured it was something like that.  “What do I have to do to get you to fold?”

I start to say “Math” but shrug my shoulders instead.


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Stuck in No Man’s Land

Last night I was at a low-stakes, private poker game marked by some very loose play.  There was frequent straddling as much as 13 BB’s, along with very light pre-flop raising (like 95s), light 3-betting, and light calling of 3-bets (with hands like T4s and 53s – the latter making quad 5’s).

It was a poker table ripe for Justin Bieber to sing, Roller Coaster, Roller Coaster.

For the most part, I was taking good advantage of this.  I bought in for 140 Big Blinds (“BBs”), felted one of the players with AQ v KQ on a Q-high flop, then felted the same player again with TT v JT after he limp/re-raised all-in with a short stack pre-flop, and built my stack to over 300 BBs.

A little later, I look down at AKo in early position, and raise to 7 BBs.  While this may seem like a large initial raise to online or casino players, it wasn’t unusual here.  My starting hand is very strong, but plays best post-flop against just one or two villains.  If I only raise to 4 BBs, everybody at the table might call and it becomes hard to know where you stand after the flop and turn.  There is one caller, then another player re-raises to 21 BBs.  For purposes of this blog post, I’ll call him “Mitch.”

Processed with VSCO with c8 preset

This is the first time I’ve played poker with Mitch.  He is a young white guy, early-to-mid 20’s, dressed like he just came from a  strange 70’s themed party – wearing what appears to be bicycling shorts, a casual shirt, old school nearly knee high basketball socks (white, with wide colorful stripes at the top of his calves), and low-cut white Chuck Taylors or similar footwear.  With a cap sporting the logo of Nag’s Head’s Lucky 12 Tavern and dark sunglasses to look like a real poker player, the whole look makes quite an impression.  I don’t really care how people look or dress, other than sometimes there are clues that help in profiling them as poker players – loose/tight or passive/aggressive or gambly or playing with a very small bankroll or whatever.  I suppose the impression here was not to be surprised by unconventionality.

Earlier in the game, Mitch had called a river all-in bluff on a very scary board (K-9-7-6-5) with KTo and won a large pot.  At the time, I was thinking that I would have folded there.  Before that, he had called a pre-flop raise from me with 92s and made a backdoor flush to beat my trip kings.

I’ve also noticed that Mitch is very friendly with another player, the one that I’ve already felted twice.  Apparently they drove to this game together and have been chatty between hands.  Mitch’s friend has a wild streak, making several large bluffs, showing his bluffs multiple times when were successful, and generally playing in a way that indicates a complete disregard for the value of his money.  Do birds of a feather flock together?

After Mitch’s 3-bet, there are two callers.  Both are very loose players who like to see lots of flops.  Perhaps both have played more than I have with Mitch and their calls indicate a certain lack of respect for his 3-bet.  I have experience with these callers, and think either of them would 4-bet here if holding a monster hand.

With all this in mind, I decide to make a sizable 4-bet myself.  With one ace and one king, I have blockers against Mitch having either of the hands I fear most – pocket aces or pocket kings.  Having started the betting from early position, I can credibly represent a monster pocket pair.  If I raise large enough, the two players who called Mitch’s bet would be forced to fold.  I make it 105 BBs.

Take that!

One player who called my original raise quickly folds.

But Mitch starts counting his entire stack.  He has 130 BBs more on top of my raise, and ships it all in.  I have him covered.

Right then it dawns on me that I’m in No Man’s Land, that terrible spot where you realize charging forward is a mistake and retreating is no good either.  In my youth I played a lot of competitive tennis.  On a tennis court, No Man’s Land refers to the area in front of the baseline, where it is difficult to make normal groundstrokes, but behind the service line, where you cannot make volleys either, at least not hitting the ball at a height that creates enough leverage to hit the ball hard or use sharp angles.  In No Man’s Land, all of your options are bad, unless you enjoy being yelled at by your tennis coach.

Another player folds.  The last player hems and haws a bit, asks for a count of Mitch’s chips, and also goes all-in, for less than Mitch’s stack.  For purposes of this blog, I’ll call him “Chuck.”  [This is really important, as I know Chuck (or whatever his real name might be) desperately covets a mention in this blog.  I hope he leaves a snarky comment after reading this.]  Chuck had about 180 BBs at the start of the hand.  His call surprised me, as noted earlier I thought my 4-bet would squeeze him out.  Even when he called, I interpreted that more as a desire to gamble over a huge pot than an indication of great strength.  Still, he could have at least one ace or king, or both, that would cancel some of my outs in the event Mitch has something like QQ or JJ.

Even so, I really don’t think Mitch has QQ or JJ.  Despite my blockers, he virtually always will have AA or KK here.  My 4-bet was so strong that his 5-bet must be stronger.  Hopefully it is KK and my ace is a live card.  Otherwise I’ll be crushed.  This is where my mistake becomes more clear.  I failed to make my 4-bet small enough to keep an exit strategy available.

Let’s review.  I’ve put 105 BBs into the pot.  Mitch has put in 235.  Chuck has put in 180.  Two other players put in 7 and 21, respectively, and later folded.  So the pot has 538 BBs in it, and it will cost me 130 more to call.  I’m getting pot odds of 4.14-to-1 to call, meaning I have to expect to win at least 19.5% of the time for calling to be mathematically, theoretically the proper thing to do.  How can I really justify folding here, even though it’s obvious that I’m in big trouble?

Some more math… heads up against Mitch, the villain I’m most worried about, if his range is AA/KK – which I consider most likely despite my blockers – and nothing else, my equity is 18.6%.  If I think he would also shove here with AKs or QQ, my equity is 33.3%.  Despite all the loose play at this game, I can only assume Mitch has a monster.  After all, loose players still get dealt monster hands just as frequently as tight players.  And not so long ago, I wrote a post entitled “Hashtag: They Always Have It.”  Mitch didn’t hesitate much before going all-in, so now I have to go with this read.

That’s heads up.  What about Chuck?  His range should be wider that Mitch’s, as his body language when calling all-in didn’t ooze great strength.  But he could have blockers to some of my outs.  Let’s give him a range of TT+, AQs+ or AK.  Against that range and Mitch’s AA/KK, my equity drops to 13.1%.  If we again widen Mitch’s range to include AKs and QQ, my equity improves to 20.4%.

Chuck could have some random suited connectors too – perhaps suspecting that Mitch and I have all the high cards and hoping for a hand like 98s to sneak to victory.  So I’ll add 98s and 87s to his range.  This, with Mitch at AA/KK would leave me with equity of 13.8%.  With the wider range for Mitch, I’m at 21.6%.  Chuck’s range has far less impact on how I stand than Mitch’s range, but if Chuck’s actual cards include any aces it kills a very important out for me.

I’m not doing all this math in my head at the table.  With Chuck’s chips in the middle now, it seems like a mandatory call.  I make the crying call, not at all happy, but would still have nearly 100 BBs left if I call here and lose.  If I had made a smaller 4-bet, say in the neighborhood of 60-75 BBs, I would have less hesitation about folding and saving my chips.

As the dealer sorts out the main pot and side pot, Chuck asks who has pocket aces.  Mitch says he has cowboys, i.e., pocket kings.  I actually feel a slight sense of relief at hearing this.  I say that I have one ace, but not two of them, and turn over my Ace of spades.  Mitch turns over his King of spades, then a red king to go along with it. Chuck doesn’t turn over either of his cards, but looks like he’s in a lot of pain.

One more bit of math:  against Mitch’s exact hand, which we now know for sure (and ignoring Chuck, since we still don’t know what he had), my equity is 30.3%.  If Chuck had folded to Mitch’s all-in bet, the pot would have been 389 BBs with 130 more for me to call.  I would need to have greater than 25.0% equity to justify calling.  While calling would be correct, it is only correct because of my betting mistake when I made such a large 4-bet that I stepped into No Man’s Land and pot-committed myself without consciously intending to do so.

If you’ve read this far, I’m pleased to report that a beautiful ace fell on the flop, and I won the whole freakin’ thing.  That’s poker I guess, and I’ve certainly been on the other side many, many times.

My stack grew a little bit more by the end of the night and I booked a very nice profit of 700+ BBs.

Thinking about Mitch’s 70’s theme appearance… That’s the way, uh-huh, uh-huh, I like it!

How Jackpots Change the Odds

I was at Maryland Live! casino near Baltimore recently, and they have some interesting bonuses and jackpots.  During this trip, they were running a special “High Hand” jackpot, where the highest hand anywhere in the poker room each hour gets a payout of $2,500.  In addition, they always have a Royal Flush Bonus, where anytime a player gets a royal flush using both hole cards, that player gets a $500 bonus and each other player at his or her table gets $100.  My friend Brian once got the table share.

So there I am, playing $2/5 no limit with about $600 on the table, and I call a pre-flop raise to $20 from the cutoff seat with Jc Tc.  The button also calls.

Flop ($60):  Ac Kc 4h

A royal flush draw.  But I’ll only get the bonus money for the Royal Flush and the High Hand if it actually hits.  What is the optimal way to play this?

The pre-flop opener checks, I check, and the button bets $30.  Opener folds.  I call.  I have a 2nd nut flush draw, gutshot straight draw, and one-outer to a Royal Flush.  So the odds look like this:

Pot:  $90

Amount to call:  $30

Let’s assume I’m behind here, and my opponent has a hand like Ax or Kx.

I’m getting 3-to-1.  I’ll hit the Royal Flush 1-in-47 times, and win $2,500 + $500 + $90 (assuming I get no further action).

I’ll hit a lesser flush 8-in-47 times, and a straight another 3-in-47 times (cannot count Qc twice), and win $90.

I’ll lose $30 the remaining 35-in-47 times, but may get another chance at the river card.

Here is the math:

1/47 x $3,090 = $65.75

11/47 x $90 = $21.05

35/47 x ($30) = ($22.35)

Add these up for Expected Value of $64.45.  That’s positive EV, so calling is correct.

Without the Royal Flush and High Hand bonuses, the Qc result is the same as any other club, so the math is:

12/47 x $90 = $23.00

35/47 x ($30) – ($22.35)

Net EV is $0.65.  Just a borderline call.

The turn card doesn’t help me, nor does it pair the board which might give the villain a full house, so the odds change only slightly.

Now I check again and he bets $45, into a pot that is now $120.  Since another card has been revealed, the denominator is now 46 instead of 47.  (We start with 52 cards.  Subtract my 2 and the 4 community cards.  The river will be one of the 46 remaining unknown cards.  Yes, it is possible that the card I want is already in the muck pile, but those cards are all part of the unknown 46.)  It will cost me $45 to try to win a pot that is now $165 including the villain’s turn bet, plus the Royal Flush and High Hand jackpots if the Qc hits.

1/46 x $3,165 = $68.80

11/46 x $165 = $39.45

35/46 x ($45) = ($34.25)

Add these up and the EV is $74.00.  Proper to call.

Without the jackpot money, it looks like this:

12/46 x $165 = $43.05

35/46 x ($45) = ($34.25)

Net EV is still positive at $8.80, so calling is still a correct play.

Lastly, we need to consider the impact if I were to bet, or check-raise on the flop or turn.  If I have any fold equity (value that I gain by winning the hand when the villain folds), how does that change the overall EV?

First of all, it must be observed that if I become the aggressor and get the villain to fold, I cannot win the Royal Flush or High Hand bonuses, a combined $3,000, WHICH I REALLY, REALLY WANT TO WIN (really, I do), as I won’t get to see another card.  But let’s do the math anyway.

Part of the challenge is that we don’t know how often he will fold.  We’ll look at 4 scenarios:

Scenario 1 – I check-raise the flop and he folds 1/3 of the time.

Scenario 2 – I check-raise the flop and he folds 2/3 of the time.

Scenario 3 – I call the flop, then check-raise the turn and he folds 1/3 of the time.

Scenario 4 – I call the flop, then check-raise the turn and he folds 2/3 of the time.

In each case, we’ll further assume that my check-raise bet size is 4x his bet on that street.

Ready for some math?

Scenario 1:

One-third of the time, he folds and I win $90.

The remaining two-thirds of the time, he calls my raise to $120.  I can win the $60 that was in the pot pre-flop plus $120 more, so…

1/3 x $90 = $30

2/3 x 1/47 x $3,180 = $45.10

2/3 x 11/47 x $180 = $28.10

2/3 x 35/47 x ($120) = ($59.55)

Total EV is $43.65.  This EV is lower than my calculation for just calling on the flop (which was $64.45), so calling is the better option – heavily influenced by the jackpots.  Take away the jackpots, and now:

1/3 x $90 = $30

2/3 x 12/47 x $180 = $30.65

2/3 x 35/47 x ($120) = ($59.55)

Total EV is $1.10.  Paltry, but better than the $0.65 EV of calling and no jackpots.

Scenario 2:

Now he is folding 2/3 of the time to my check-raise to $120 (i.e., 4x his bet of $30):

2/3 x $90 = $60

1/3 x 1/47 x $3,180 = $22.55

1/3 x 11/47 x $180 = $14.05

1/3 x 35/47 x ($120) = ($29.80)

Net EV is now $66.80.  Whoa Nelly!  Even the the jackpots that I forego when I can make him fold, the EV is now higher than the EV of calling and chasing the jackpots.  If (and it is a big IF) this villain would really fold as much as 2/3 of the time to a 4x check-raise, that becomes a better play than calling and chasing. Of course, we don’t know what he has, nor have we played with this particular villain for very long.

Scenario 3:

I call the flop.  Now the pot is $120, I check, he bets $45 and I check-raise 4x to $180.  One-third of the time, he folds and I win $165.

The remaining two-thirds of the time, he calls my raise to $180.  If I hit one of my outs on the river, I can win the $120 that was in the pot after the flop betting plus $180 more, so…

1/3 x $165 = $55.00

2/3 x 1/46 x $3,300 = $47.85

2/3 x 11/46 x $300 = $47.85

2/3 x 35/46 x ($180) = ($91.30)

Total EV is $59.40.  This EV is also lower than my calculation for just calling on the turn (which was $74.00), so calling is still the better option due to the huge jackpots.  Take away the jackpots, and now:

1/3 x $165 = $55.00

2/3 x 12/46 x $300 = $52.15

2/3 x 35/46 x ($180) = ($91.30)

Total EV is $15.85.  Again this is slightly better than the $8.80 EV of calling on the turn with no jackpots.

Scenario 4:

Now he is folding 2/3 of the time to my turn check-raise to $180 (i.e., 4x his bet of $45):

2/3 x $165 = $110.00

1/3 x 1/46 x $3,300 = $23.90

1/3 x 11/46 x $300 = $23.90

1/3 x 35/46 x ($180) = ($45.65)

Net EV is now $112.15.  Once again, with the greater fold equity, this becomes a better play than chasing the jackpots.

So what finally happened?  Of course, the river was a total brick and we both checked.  He had AQ, including the Qc which was my gin card, and wins the pot.

Given what we now know about his hand, how often would a typical, regular casino $2/5 no limit Holdem player fold to a flop or turn check-raise?  Obviously I’m putting a lot more chips in there, and we now know the only way I could have won that pot was to make him fold.  But would he fold?  If we repeated this hand 100 times, would he fold often enough for my aggression to be profitable?


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