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Combinatorics – Monotone Flops
Combinatorics is a powerful in poker for helping you analyze flop texture.
Monotone Board
— if the board is all the same suit, (AKQ all hearts)
1) how many combos are made flushes?
There are 10 other hearts (J – deuce). To make a flush, you need two of them in your hand. Pocket pairs don’t contribute since they are not the same suit. To do the math, I use the counting rule (M ways x N ways) or in this case (10 heart’s on 1st card x 9 hearts on 2nd card) making 90 combos to flop a flush.
2) how many combos are flush draws?
Again, there are 10 heart’s and you can only have 1 heart in your hand to have a flush draw. This means, the previous 90 combos are excluded. Instead, we need to assume that you have 10 ways to be dealt the heart needed to qualify as “having a flush draw” however, you need to be dealt a non-heart on the 2nd card. There are 100 possible hands between J and deuce. Note, 10 of these combos are pocket pairs and 90 non-pocket pairs.
For those 90 non-pairs, each has 16 combos total but we care only about the ones with a single heart.
If we assume you have non-pocket pair like JT with one heart, we have these combos (Jh w/ Ts, Td, Tc) and (Th w/ Js, Jd, Jc). This means only 6 combos per non-pocket pair. A total of (90*6=540) combos for the range between J-2.
If we assume you have a pocket pair like JJ w/ 1 heart then there are only 3 combos for each pocket pair (Jh w/ Js, Jd, Jc). This is a total of (10*3=30) combos of any pocket pair with a flush draw.
Recap
A total of 90 combos of starting hands can flop a flush on a monotone flop.
A total of 570 combos of starting hands that could have a flush draw on a monotone flop. You can filter the range if you assume they will fold some of these flush draws. Eliminate 3 combos for each pocket pair they fold (like the 4h4s). Eliminate 6 combos for each non-pair they fold (like the 2h3c).
There are 69 possible starting hands (13×13) where 13 of them are pocket pairs AA-22 and rest are non-pairs. For each pocket pair, there are 6 combos. For each non-pairs, there are 16 combos where 4 of them are suited and 12 are offsuit. This means that the starting combinations for pocket pairs are (13*6 = 78) and non-pairs are (78*16=1248) for a total of 1326 combos of starting hands.
Let’s do some math with an example.
If we are up against Gus Hanson who will play any two cards, the flop is AKQ all hearts. He check raises us on the flop. We know that he always has at least one heart when he reraises here. Never raising a worse 2 pair, not raising any flopped sets, not raising a straight without a heart.
— how often does he have the flush?
- 90 combos of 1326 total can flop a flush on a monotone flop. This is 6.78%.
— how often is he semi-bluffing?
- 570 combos of 1326 can flop a flush draw on a monotone flop. This is 42.98%.
This means that Gus Hanson can check raise up to (~49%) with mostly bluffs on any monotone flop he sees.
— what is AK’s equity against this range?
AK would need to hit a full house if up against a made flush. AK has 4 outs twice to hit or 16% to hit it by the river.
Against a semi-bluff, my opponent has 9 remaining hearts to hit because his hand has 1 heart and 3 hearts are on the flop. Again, the rule of 4 and 2 estimates that these 9 outs have about 36% equity to hit the flush by the river.
Analysis
16% to win against 6.78% of range when opponent has a made flush.
64% to win against 42.98% of range when the opponent has a flush draw.
Should you raise, call, or fold AK?
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10 Replies
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Typo — 169 starting hands, not 69.
Also, I want to add this
Flop Textures:
- Monotone flops occur about 5% of the time
- Two-Suited flops occur about 55% of the time
- Rainbow flops occur about 39% of the time
- Flops w/ No Pairs occur about 82% of the time
- Flops w/ One Pair occur about 17% of the time
- Flops w/ Trips occur about < 1% of the time
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Dude this is awesome but I’m going to need some people to talk through it with… like, some sort of “community”… hmmm where could that be? Seriously though, I’m excited to see what others think and how we can use this information.
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This also assumes that V is playing every suited heart combo, which for most players is a stretch. Maybe not for Gus Hanson. Against a typical villain, AKQ all hearts is far better than say 456 all hearts because it removes a lot of the suited hands they get here with. Still, even if we give them all those heart combos, then we have to give them all the spade, club, and diamond combos too. But we’ve just said they wouldn’t check raise those, so given our range read, we’ve lost a lot of the hands we want V to have (complete air, worse two pairs, pairs and straight draws etc.). The unsuited hands are even rougher to find that get here. J-10, J-9, or 10-9 off with hearts?
So although in straight combos we rate to be ahead of any hand with at least one heart. If we can narrow that range down, I think we’re only seeing this raise with a hand like AxJh Ax10h Jx10h Jh10x and or hearts. I don’t think we’re being check raise by Kx/Qx with J/10h
I’m calling this once because I do have outs and because I may still be ahead, but this about the bottom of my calling range with this read and I’m likely folding to much further aggression on blanks.
Absent the read described, I think many villains are very capable of check raising here without a heart. And that makes the call a little more easy. Those are my two cents.
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Analysis
16% to win against 6.78% of range when opponent has a made flush.
64% to win against 42.98% of range when the opponent has a flush draw.
??% to win against calling range, assuming he doesn’t re-raise the flop
100% to win against folding range
Should you raise, call, or fold AK?
Let’s put some numbers to this hand,
You both start the hand with $750 after pre-flop action. You have AK on an AhKhQh flop, your opponent checks, you bet $25 into a pre-flop pot of $50, and your opponent raises to $100 more.
- You can fold. Stack size would be $725. Losing a bet of $25.
- You can call. You’ve invested $125 with 2 more streets to go and only $625 left in your stack. The Pot Size would be $300.
- You can raise. You would be raising an additional $625 and offering pot odds of ((625 + 300)/(625)) or about 1.5 to 1.0. This has a break-even of 40% win needed for them to call the raise and profit from you.
- Which of the villains hands will call our raise?
- made flushes
- this is only 6.78% of the starting hand range
- we win 16% against this range. Bad Shape 🙁
- flush draws
- It’s fair to say that not all draws will call. Let’s assume that maybe 1/3 or the (Q, J, T) high flush draws will call and the others (9, 8, 7, 6, 5, 4, 3, 2) high flush draws will fold to a re-raise. This would be 1/3 of 42.98% or about 14% of the starting hand range.
- we win 64% against this range. Good Shape!!!
- two pairs
- same hand (AK)
- there are 4 combos of AK possible to chop with us on the AKQ flop when you have AK. “Everyone loves a chopped pot”
- worse two pairs (AQ, KQ)
- There are 6 combos of AQ and 6 combos of KQ possible when you have AK on the AKQ flop.
- we win 92% against this range, assuming they don’t hit the two outer.
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If we raise, our opponent might call with a variety of hands and I want to know my equity against the made hands (straight, set, flush, two pair) and the bluffing range (flush draws, gut-shots).
— AK vs JhTc — 17% vs.82%
- our opponent has a flopped straight with the nut flush draw. We have 4 outs.
— AK vs QQ — 17% vs 77%
- our opponent has bottom set of queens. Neither hand has a flush draw. We have 4 outs. There is a ~6% chance of a tie due to running hearts.
— AK vs 3h2h — 17% vs 80%
- our opponent has the worst possible flush. We have 4 outs.
— AK vs KQ — 86% vs 8%
- our opponent has a weaker two-pair. They have only 2 outs to suck out. There is a ~6% chance of a tie due to running hearts.
— AK vs Jh2c — 57% vs 42%
- our opponent has the nut flush draw and a gut-shot. They have 12 outs to win. The 9 remaining hearts to hit the flush and the 3 remaining Tens to hit the gut-shot straight.
Expected Value
<div>
</div>Fold EV
= (% opponent folds)*($300)
Call EV
= (% opponent calls) * [(% hero win)($300 + $625) + (% hero lose)(-$625)]
Total EV
= Fold EV + Call EV
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To do an example, let’s make assumptions:
- our opponent will call the raise 25% of the time and fold 75% of the time
- If we win ~20% of the time, what is our expected value?
Fold EV
= (75%)($300)
= $225
Call EV
= (25%)[(20%)($925) + (80%)(-$625)]
= (0.25)[(0.20)(925) – (0.80)(625)]
= (0.25)[185 – 500]
= $-78.75
Total EV
= 225 – 78.75 = $146.25
For the price of calling $625 more, our opponent would need to call us more often (higher than 25%) to reduce our fold equity. We are making a lot of $$$ every time our raise forces a fold. To do this, our opponent will need to start calling with a slightly weaker range of hands (drawing hands).
Let’s adjust our assumptions:
- our opponent will now call 50% of the time and fold 50% of the time
- since they are drawing more often, I estimate that we win ~30% of the time and lose 70% of the time, what is our expected value?
Fold EV
= (50%)($300)
= $150
Call EV
= (50%)[(30%)($925) + (70%)(-$625)]
= (0.50)[(0.30)(925) – (0.70)(625)]
= (0.50)[277.5 – 437.5]
= -$58.75
Total EV
= $150 – $58.75 = $91.25
By calling more often, our opponent has changed the EV in their favor. Our expected value decreased from $146 to $91.
Now, let’s get fancy.
What % win is needed to break-even when our opponent calls 50% of the time?
Fold EV + Call EV = 0
Fold EV
= (0.50)(300)
Call EV
= (0.50)[(W)(925) + (L)(-625)] where L = 1 – W
= (0.50)[(W)(925) – (1 – W)(625)]
= (0.50)[(W)(925) – (625 – 625W)
= (0.50)(925W + 625W – 625)
= (0.50)(1550W – 625)
Total EV
(0.50)(300) + (0.50)(1550W – 625) = 0
150 + 775W – 312.5 = 0
775W = 162.5
W = 0.209677 or 20.9% win is needed
This means, if our opponent calls us 50% of the time, we only need to win 20.9% of the time to break-even.
- If we win more than 20.9%, our EV increases.
- If our opponent calls less often than 50%, our EV increases.
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What % win is needed to break-even when our opponent calls only 25% of the time?
Fold EV + Call EV = 0
Fold EV
= (0.75)(300)
Call EV
= (0.25)[(W)(925) + (L)(-625)] where L = 1 – W
= (0.25)[(W)(925) – (1 – W)(625)]
= (0.25)[(W)(925) – (625 – 625W)
= (0.25)(925W + 625W – 625)
= (0.25)(1550W – 625)
Total EV
(0.75)(300) + (0.25)(1550W – 625) = 0
225 + 387.5W – 156.25 = 0
387.5W = 68.75
W = 0.1774 or 17.7% win is needed
When they fold more often, we need to win less often.
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Use this applet to play with the numbers for Poker EV.
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There’s some great technical application here folks, come and take a gander! @ARW when you are playing in real time, what portion of this process do you have internalized, what portion do you use aids to recall in real time, and what portion do you typically abandon for real-time play and instead apply during pre- or post- play study or review sessions?
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<b style=”font-family: inherit; font-size: inherit;”>what portion of this process do you have internalized?
- Most of the math is done in my head. I’m estimating and mapping it out. I try not to make assumptions early. Good assumptions are based on making good observations and watching the action.
- In Limit poker, or “no foldem holdem” as some say, combinatorics has helped me quickly evaluate my hand to find (good cards, bad cards) on the turn and river. I play a lot of multi-way hands, I see a lot of flops, I need to be aggressive to win, and most importantly, I need to have the best hand on the river when I always get called.
what portion do you use aids to recall in real time?
- In my spare time, I write code and phone apps. When I step away from the table, I occasionally use a custom phone app to double check a hand or look something up.
what portion do you typically abandon for real-time play and instead apply during pre- or post- play study or review sessions?
- At this point, I don’t really calculate pot odds anymore except on paper because I’ve played enough hands to have a decent estimate in real-time.
- After playing a hand, I try to think it through with an analytical lens and try to avoid thinking about how the result was bad.
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Just re-upping this amazing post so people can benefit from the analysis from @ARW and see how working on this stuff off the felt can help you to internalize it so your mind can focus on other factors when you are in-game.
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