KKing David

Ruminations on poker

Archive for the tag “combinatorics”

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…

Math

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.

Combinatorics

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.

Frequencies

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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Hashtag: They Always Have It

I just returned from a weekend poker trip to New Orleans.  There were eight guys on this trip, one of whom – for purposes of this blog I’ll refer to him as “Tony” – was celebrating a birthday during the weekend.

Happy Birthday Tony!

We stayed in some fancy digs in the French Quarter, and played a lot of poker at Harrah’s New Orleans casino.  On Saturday, there were a number of interesting hands, all at blinds of $1/3.  Because this poker room allows players to buy-in up to the largest stack on the table, and also lets players straddle $6 from any position, the game plays larger than most $1/3 games, and much larger than most $1/2 games.  Let’s get started.

Hand #1 – Nut Flush on a Paired Board

This was early in my day, so I didn’t have much in the way of player-specific reads.  I have Kd Js on the button, and raise to $15 over a single limper.  I started the hand with approx. $300 and the BB is the main villain and has me covered.  The SB folds, but BB raises to $30.  He could simply be putting me on a wide range trying to leverage my position on the button, and trying to re-steal, but his raise is too small if his goal is to make me fold.  He is a middle aged white guy (like me!) making a barely minimum raise, so I need to give him credit for a strong or very strong hand.  Everyone else folds, and I call.  Heads up, in position, I’m willing to call $15 more with $52 already in the pot.  

Flop ($67):  Ad Jd 2d.  All diamonds.  He checks.  I have the King of diamonds for a nut flush draw, and middle pair.  While I could bet here as a semi-bluff, I actually have some showdown value and decide to take the free card instead.  I’m not sure he will fold any Ace to a bet.  I check.  

Turn ($67):  2c.  This card pairs the board but doesn’t otherwise change anything.  He checks again.  Does have have KK or QQ and hoping to get to a cheap showdown?  In that case, perhaps I should make at least one steal attempt, but I’m not sure he would fold, especially if he has Qd Qx.  Free cards are good, right?  So I check again.

River:  ($67) 8d.  Awesome.  This completes my nut flush.  I’m anticipating another check when he fires out a bet of $45.  I should have the best hand here, and might get a crying call if he has Ax Qd or Qd Qx.  There really isn’t much else in his pre-flop 3-betting range that can make a weaker flush here, and he wouldn’t call me without either that or better.  The Ad and Jd are on the board, and I have the Kd.  Could he be as weak as Td Tc?  I can’t imagine anything else will call a raise.  If he comes back over the top of my raise, however, I need to be concerned about AA or JJ, although I would have expected either one of these to make a small be on the turn, for value, since he would have turned a full house and would want to continue building the pot.  

I raise to $110.  With very little hesitation, he declares that he’s all-in.  Dammit!  Now I pissed at myself for raising instead of just calling, although I can hear the voices of those who advocate bet/folding and raise/folding for value as superior long-term value strategies over calling whenever you have a big, but non-nut hand.  I also here the voice of one of my favorite poker podcasters saying “hashtag: they always have it” (#theyalwayshaveit).  

While trying to get emotionally unattached from my nut flush and decide if I’m a bad enough player to donate the rest of my chips, I replay the hand, try to recall any observations about this specific player and ponder his body language when he went all-in and the meaning of his very small pre-flop 3-bet.  He seemed confident, not the least bit afraid of the fact that he cannot have the Kd.  Yes, he has to have pocket aces here.

I fold, and in an act of pure kindness that I’ll exploit if given the chance, he flashes his AA for the entire table to see.

In hindsight, I think raising was a mistake on that river, as there were too few hands that would call a raise, and his pre-flop action certainly suggested AA as part of his range, with a small raise begging to be called.

Hand #2 – Does Small Turn Bet Signal Weakness?

After Hand #1, I add another $200 to my stack.  Shortly after that, I am dealt Kh Jd in middle position, and two players limp in front of me.  What is it with “jack-king-off” and me this weekend?  I felted a woman (felted her, not felt her!) for a nearly $400 pot with this holding late the previous night, and here it is again in both Hand #1 and #2.  This time, I raise to $18 and get called by the button and one of the limpers.  I started this hand with $326 and the main villain on the button has me easily covered.

Flop ($61):  Tc 9s 7c.  This is an interesting flop, very drawy, giving me a double gutshot straight draw in addition to two over cards, while also hitting a lot of pre-flop calling ranges for flush draws, flopped straights or straight draws, combo draws, sets and more.  The first guy checks.  I want to retain the initiative, and bet $35 as a semi-bluff.  Hopefully they missed this flop badly and will fold.  Or the button will fold and I’ll have position on the other guy.  The button does not cooperate, instead raising to $85, and the other guy folds leaving us heads up, with me being out-of-position

I don’t have any reads or tells on this player, as he is new at this table, a white male who appears to be in his mid-to-late 30’s.  Since he called a raise pre-flop on the button, his range can be pretty wide but would exclude 3-betting hands like AA, KK or QQ.  For now, I’ll give him credit for possibly raising with a flush or straight draw, combo draws, and strong made hands of top pair / top kicker or better.  This range is JJ-77, Ac 2c+, Kc Jc+, Qc Jc, JT, J8s, T9, T7s, 98, 97s, 87, 86s, ATo.  

(Or:  pocket pairs from 77 through JJ, Ac along with any other club, Kc along with the Qc or Jc, Qc Jc, Ace-Ten off-suit, any JT, T9, 98 or 87, and any suited J8, T7, 97 or 86.)

The overall player pool in this poker room is very loose pre-flop.  In position, a large portion will call with virtually any connecting cards (like JT, T9, 98 or 87) or suited 1- and 2-gappers (like J8, T8, 97 or 86). 

This range includes 139 different combinations of two cards, more than 10% of all possible combinations.  Even without a pair, I have 32.7% equity against his range.  With $180 already in the pot and it costing me $50 more to call, I only need 21.7% equity to call even if I’m only assured of seeing one card.  If it’s a blank and he fires out a large turn bet, I might have to bail out.  I call.

Turn ($230): Ks.  Now I have top pair, plus the double gutshot draw, and a second flush draw is now also a possibility.

I check and the button bets $50, less than 1/4 of the pot.  This is a strangely small bet.  With so many draws possible, I would expect a much larger bet from his stronger made hands – flopped straights or sets – as there would be a lot of hands I could have that would call again and he should be protecting against the draws.  If he were semi-bluffing with a flush draw, he should be betting much more here, or else just check back and take a free card.  My range is narrower than his, as I was the pre-flop raiser and less likely to be raising with the weaker connectors and gappers that he may have called with favorable position.

Against his full range, my equity is now 60.5%.  I’m beating hands like T8, 98, 88, 87, AT, JT, and have more outs against his 2-pair hands.  Smelling weakness, I check-raise all-in for $173 on top of his bet.  Now there is $503 in the pot and it costs him $173 to call.  I wish my stack were deeper, but this is still a large enough bet to make him pause and think.  

He finally makes what looks like crying call and I flip my cards over right away.  The river is another 7, putting a pair on the board.   He turns over 99, for a flopped set & rivered full house to scoop up the pot.  I must have sold him that I was representing QJ there for a turned straight and he felt like he was gambling with the call, whereas in reality all he needed was to fade a Q or 8.  

His hand was within the range I assigned to him, but right up near the top of that range.  Once again, #theyalwayshaveit.  I still think his turn bet was curiously small on such a wet board.  Maybe he’s just not a thinking poker player, and that caused a misread on my part.  Against his exact hand, my equity was 25.3% on the flop and 18.2% on the turn.

At this point of my day, I’ve been playing about 45 minutes blasted away $500.  I take a break for a few minutes, then move to another table where I buy-in for $500 more.

Hand #3 – Donk Bets on Flop Signal Weakness

The villain in this hand is an older (early 70’s?), white haired guy, somewhat the “old man coffee” (“OMC”) type — enjoying his poker game, passing time playing fairly tight and straightforwardly.  He started this hand with about $350 in chips and I cover.  After he and a couple others limp in, I raise to $20 with QQ.  Two others call, then OMC calls.  

Flop ($83):  J 9 4 rainbow.  It checks to OMC, who bets $50.  With straightforward players, this “donk bet” (betting into the pre-flop raiser) is usually a good but not great made hand.  This looks exactly like he has AJ.  With 2-pair or better, he would be more likely to try for a check-raise, with greater confidence that his hand would still be best on the turn if it checks all around.  If he does have AJ, I may be able to get his entire stack if I’m patient.  Not too quickly, I call, and the others all fold.

Turn ($183):  7c now puts two clubs on the board.  OMC bets $25, a strangely small bet, but this doesn’t change my read that I should target him having AJ (despite my misread of a small turn bet in Hand #2, it would be an example of recency bias to change my read here).  I raise to $125 by tossing out five green chips.  I hope this somehow looks bluffy. He tanks, fidgets, squirms, appears to decide to fold and then change his mind.  OMC’s as a general rule are stubborn and hate folding top pair / top kicker hands when they’ve been sitting for a long time just to get one.  He calls.

River ($333):  4s.  This is a great card for me, pairing the bottom card on the board.  If somehow he had donk bet with top two pair (J9) he’s now counterfeited.  He checks, I slide out $200 to be sure I cover his remaining ~ $150 or so.  He says “you’re going to put me all in aren’t you?”  And calls with an expression on his face that says “I know I’m beat, but I’ve gone too far in this hand to give up now.”  Without waiting for me to show first, he flips over KJ, confirming my read was almost exactly right.  My QQ is good and I drag home a large pot.

In this case, #theyalwayshaveit turned out to be good for me.  He had it, but “it” was a hand that I beat and the straightforward nature of his play enabled me to confidently play for stacks with a 1-pair hand.

Hand #4 – Can I Fold KK Pre-Flop?

After the previous hand, OMC left the table and was replaced by another older guy (late 60’s?) who was more of an “old man beer” (“OMB”) than “old man coffee.”  He was more social than the first guy, but hadn’t been playing long enough for any other reads prior to this hand.  At the start of this hand he has about $250-300 and I have him covered.

After OMB limps in under the gun and the next player folds, I look down at Kh Kd and raise to $17.  A woman calls all-in for her last $11 and there is one other call.  Then OMB re-raises to $85.  WTF?

Wait… did he limp in earlier or was he the big blind?  Just to be absolutely sure, I confirm the position of the button on the table as it is in the corner opposite my seat.  Then carefully count small blind, then big blind, then OMB.  Yes, he definitely limped then re-raised.  Dammit!  People love to do this with pocket aces.  

I ponder Dan Harrington’s seminal books on cash games, where he says no one is good enough to put an opponent on exactly AA and be right often enough to justify folding pocket kings.  Since you can re-load anytime in a cash game, his philosophy is to just take your lumps when this happens (unless playing with very deep stacks), so I consider just closing my eyes and shoving all-in.

Then I ponder Occam’s Razor, a problem solving principle that says when there are multiple explanations for an occurrence, the simplest one is usually better.  That’s a fancy way of saying #theyalwayshaveit, the simplest explanation for large bets.

And I ponder former poker coach Bill Hubbard, who constantly asked, based on what we know about a villain, “what does he have and how will he play it?”  Working backwards, how did he play it (limp / re-raise) and what does that mean he has?  #theyalwayshaveit, AA in this case.

And I ponder myself.  I have to be a good enough poker player to fold here.  But it’s a major ego destroyer.  I mean, like, c’mon, this is KKing David’s poker blog.  I’m KKing David.  I have KK here.  This pot is supposed to be mine.  After getting beat up in Hands #1 and #2, this is the kind of hand I need to get un-stuck.

This time, I win the battle with my emotions, and reluctantly make the disciplined fold.  Since one player is all-in for $11, we get to see the runout.  She has 99 and, sure enough, OMB has AA.  There are no aces, nor kings, nor nines on the board and OMB scoops up a modest pot.  

Deep breathing exercises…

Hand #5 – He Can Only Have Pocket Queens (or Quad Jacks???)

A new player has recently joined the table, two seats to my left.  The entire time I’ve been at this table, one of the other guys on this road trip has been on my immediate right.  For purposes of this blog, I’ll call him “Zach.”  Shortly after the new player arrived, Zach softly commented to me to beware of this new player, who seems to be a very strong and aggressive player, although he is drinking a beer and it’s 3:00 on a Saturday afternoon.  I notice his ability to leverage favorable position against passive players.  

In this hand, he is the main villain in the cutoff seat, with a stack around $550, which I cover.  The button straddles for $6 and three players call.  With my first AA of the day, I make it $35.  The villain tanks, counting and re-stacking his  chips in a manner that tells me he is thinking about re-raising.  Zach told me later that it took so long for this guy to act that he was ready to call the clock on him.  I just wanted to see him re-raise, but ultimately he calls $35.  The button folds, but there are two more out-of-position callers.

Flop ($150):  J high and very dry.  I don’t recall exactly the two lower cards, other than there was no flush draw nor possible made straight.  It is checked to me, and I bet $100.  The villain calls almost immediately.

His range for 3-betting pre-flop might be TT+, AQ+ absent any other reads.  This is important, as his pondering a 3-bet indicates something in the weaker part of this range or slightly weaker, and also rules out – assuming my read is correct – weaker calling hands.  While the player pool here in New Orleans is very loose pre-flop, there hasn’t been much light 3-betting pre-flop, and the straddle and callers in front of me has already push the bet-sizing above normal ranges.   He would definitely 3-bet with AA or KK, so that leaves his pre-flop calling range as 88-QQ, AJ+.  I heavily discount the TT, 99, 88, AJ part of this range as I don’t think he would take so long to decide to just call my $35 bet.  

Now he has confidently called $100 on a J-high flop.  There is no T on the board, so he cannot have two over cards with a straight draw (AK, AQ, KQ).  We can rule those hands out.  He would have to think a bit before calling with TT or any lower pocket pair, so we can rule those out also.  I’ve already ruled out AJ based on the pre-flop read, so that leaves QQ and JJ and nothing else.  If he flopped a set of Jacks here, it makes sense for him to just call and see if either of the other players wants to come along too, but they both folded.  There are six combinations of QQ and three combinations of JJ.  

Turn ($350):  A low brick arrives that really changes nothing.  I check, partly for pot control as my hand is still only one pair and I don’t want to play for stacks yet, and partly to see what he does.  He checks  back.

River ($350):  Another J.  Ugh!!!!!  I check.  At first, I hate this card, as an instinctive reaction to seeing the top card on the board pair.  What if he has AJ or KJ – even though I previously ruled these out?  

He bets $200.  Gulp!  I take a deep breath and re-play the hand and re-evaluate my reads based on the action on each street.  I have to trust my reads!  There are still six combos of QQ that he can have, but now only one combo of JJ.  #theyalwayshaveit… in this case “it” is either QQ 85.7% of the time, or JJ 14.3% of the time.  It is an unusually precise read.  Notice the contrast between this read and Hand #2 above where he could have over 130 combinations.  I’m confident that his pre-flop chip handling signaled thoughts about re-raising.  Perhaps I should check-raise all-in here, which would really be sick, but that seems very unwise.  I call and he shows QQ.  Cha-ching!

Zach later pointed out, and I agree, that his river bet there was terrible.  The villain was in a “way ahead / way behind” situation.  The weakest hand I could would beat him is KK, which will almost never fold there.  Most players with a better hand won’t think as much as I did (a flaw? or a gift?) and would just call.  And the strongest hand that he beats would be TT, which will almost never call.  If you can’t get a better hand to fold nor a weaker hand to call, and you are last to act, just check back and go to showdown.  I’m not sure if he thought he was betting for value or if he knew he was turning his QQ into a bluff, or if he even knew why he was betting on that river.

I drag home another large pot, just glad to have turned my day around.

Too Nitty?

I may have been too nitty on this hand last night.  When this happened, I was having a bad night, having tried to turn KK into a bluff and having this backfire badly, then having QQ cracked by JJ after a villain called a large pre-flop 4-bet (and then admitted that he thought his JJ’s were in trouble).

The good news, perhaps, is that I’m aware things are going badly and trying to be careful not to let emotions overrule good decisions.

Now I have KQo in middle position.  This being a $1/2 no limit Hold’em game, I raise to $11 following one limper, which is fairly typical for me.  The button calls (I’ll call him “Mike”), one of the blinds calls, and the limper calls.

Flop ($40):  Kd 8d 7s

This looks like a good flop for me, but the flush and straight draws can hit the villains’ ranges.  It is checked to me and I bet $17.  In hindsight, this bet is too small, and should have been more than 1/2 of the pot.  But my TPGK, while good, is still just a 1-pair hand and I don’t want to bloat this pot out of control.

Mike, on the button, raise to $43, and both other players fold.  At the start of this hand, I had about $250, and Mike has me covered.

Mike is a tough player, good at hand reading and willing to bluff.  And he has position on me.  With many villains, my default in this situation is to give him credit for 2-pair plus most of the time, and top pair / over pair the rest of the time.  Let’s quickly eliminate the only over pair… AA, as he didn’t 3-bet pre-flop.  But I know Mike will play his draws aggressively, and he might have a K as well.

I decide to re-raise, $45 more.  I want to leave myself room to fold if he comes back over the top and convinces me that he’s flopped a set.  I also, however, want to find out where I am in this hand (and believe I might be good), as I realize after his raise that my C-bet was too weak.  If he calls, that suggests he is on a draw and I can re-evaluate after seeing the turn card.

Mike tanks for a bit, and seems to be considering multiple options.  Then he announces “I’m all in.”

Uh-oh.

Now I go into the tank.  The pot is indeed bloated, now approximately $220, and will cost me $150 more to call.  My hand is just one pair, and I hate going broke with just one pair (even moreso having already been felted once in this session).

I settle on this range:  88 (3 combinations), 77 (3 combos), multi-way draws (Td 9d, Jd Td, 7d Ad, 7d 6d, perhaps 7d 9d, perhaps Ad Jd and Ad Td — total of 7 combos).  I note the K is a diamond, so a top pair / flush draw cannot be in his range.  I think he would 3-bet pre-flop with Ad Qd and any AK.  Neither of my cards is a diamond.

With the sets, he has me crushed.  With the biggest draw… Td 9d, he has 15 outs and would be a favorite to win the hand.  My equity doesn’t have to be over 50% to justify calling from a math standpoint, as I would be calling $150 to win $370.  I really only need 29% equity ($150 / $520) for calling to be correct.  But I’m not really thinking about that at the table, I’m thinking about being stuck 2 full buy-ins if I lose, and that thought pisses me off.  And I’m thinking about my long-term image if I’m viewed as the guy who over plays a one pair hand (especially with less than top kicker).  The draws seem more likely than the sets, as I’m not sure he shoves a set here.  On the other hand, with a set he might think I’m overplaying AK or AA here, and shoving a set for value could be smart before the board gets scarier.

Reluctantly, I fold.  This wasn’t a math fold, it was a variance fold.  I’d already suffered more variance (see above re QQ losing a large pot to JJ) in this session than I like, wasn’t prepared to buy-in again if I went all-in and lost here, and was sitting on the left of a couple of players who have a history of spewing off lots of chips.  I like my chances if I keep playing.

After saying I thought he had something like Td 9d, he says I’m close and some prodding gets Mike to tell me he had a 7d and another diamond, for bottom pair and a flush draw (14 outs).

Against the range above, my equity is 27.2%.  Against only Td 9d, the drawing hand with the most outs, my equity is 43%.  Against 7Xdd, my equity is 47%.  Gambling for sure, but high enough to call.

Methinks I should have called.  We’ll never know.  But given how poorly the rest of the session went,  I might as well have gambled here.

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