We have preflop range of lets say A2+, 22+
On average we want to bet 10% of hands on the flop from this range (best 10% by equity vs random hand on the flops)

The question is: how to calculate what hands to bet on each flop, so that finally we have 10% bets on average.

So far I'm doing this, on the flop calculate the equity of my range (A2+, 22+) vs rando hand and bet the top 10% of the range for this flop.
This is obviously not good because we often include hands that we can skip and compensate with much stronger hands on other flops to achieve our final 10%

Please note this is pure poker math question, I'm not interested in player tendencies, implied odds, meta game and so on.

So you want a static cbetting range that is 10% of your opening range no matter what the flop is?

Otherwise I'm not understanding your question. Also taking just the top 10% equity hands in your range is quickly going to net you less EV as opponents figure out you're never bluffing or bluffing with the best draw.

I want my average betting range to be 10%
To be more precise I want to know how to calculate such betting range for each flop that these combined net to 10% average for all flops

Also note the range and % are just an example
Also I dont mind if I can calculate it as somewhere about 10% I don't need this to be 10% exactly
Finally I will be glad for ideas of calculating, that just improve the algo in my first post, dont need this to be the uber "best" algo


> is quickly going to net you less EV as opponents
I need to calculate this first, later I will focus on opponents and such.

Well my math could be a little off here but that's never stopped me from posting before so here we go. Sorry in advance for long post.

Your range preflop is A2+ 22+ so the total number of combinations in your range is 270 according to my calculations.

I got this by calculating the number of unpaired hand combinations (4×4 = 16) and multiplying by the number of different ranks that go with your A (2-K = 12 ranks) so 16×12 = 192. Now there are 6 combinations for every pair (3×4/2×1 = 12/2 = 6) and 13 total ranks that can make a pair (2-A = 13 ranks) so 6×13= 78 pair combinations. 192+78 = 270 total range combinations for you.

So if we assume you take 270 combinations to the flop you'd bet 27 of those combinations to bet 10% of your range.

However, the flop cards will lower the number of possible combinations in your range. Lets look at some examples:

3 different cards (no A):

Unpaired combo effect: the 3 ranks on board will have 1/4 of their combinations removed. 16×(1/4) = 4 combos × 3 ranks = 12 combos total.

Paired combos effect: Pair combos remaining for each of the 3 ranks are 3×2/2×1 = 6/2 = 3. 6 original combos - 3 combos remaining = 3 combos missing. 3 combos missing × 3 different flop ranks = 9 pair combos missing.

21 total fewer combos for a range of 249 combos and a betting range of 24.9 combos to maintain 10%

3 different cards with an A:

Unpaired combos effect: 1/4 of ALL our unpaired hands are missing from our range because of the A. 16/4 = 4 combos per rank × 12 ranks = 48 missing combos. An additional 2 ranked hands will be missing another 1/4 of the remaining 12 combos for that rank due to the 2 other rank cards on the flop. 12/4 = 3 combos missing × 2 rank cards = 6 additional missing combos. 48+6 = 54 unpaired combos missing from our range.

Pair combos lowered: Remains the same so 3×3 = 9

63 fewer combos in our range. 207 combos going to the flop and bet 20.7 combos to maintain 10%.

Paired flops (no A).

Unpaired combos effect: For the paired rank, 1/2 our combos will be missing for that 1 Ax rank. 16/2 = 8. 1/4 of the Ax hands for the other rank card on board will be missing from our range. 16/4 = 4 . 8 + 4 = 12 combos missing from our range.

Paired combos affected = 5 fewer pair combos for the pair showing on board. 3 fewer combos for the other card. 8 combos total.

20 fewer combos in range for 250 total combos on the flop. We bet 25 hands to maintain 10%.

Paired board (with A, A is not paired):

Unpaired combos affected: 1/4 of 12 Ax ranks will be missing due to A on flop. 16/4 = 4 × 12 = 48. Another 1/2 of the paired rank will be missing from the 12 combos that remain. 12/2= 6. 6+48 = 54

Paired combos affected:
Same as above 8 combos, 5 for the pair ranked cards and 3 for the A.

62 total combos for 208 combos in range and betting 20.8 combos to maintain 10%.

Paired, with A on board and A is paired:

Unpaired cimboss effect: 2 aces lowers unpaired hand combos by 1/2 192/2 = 96 missing combos. 1/4 of the remaining Ax hands with the rank of the card on board will also be missing from our range. 16/2 = 8. 8/4 = 2. 96+2 = 98 total unpaired combos missing from our range.

Paired combos effect: Same 8, 5 for paired cards and 3 for unpaired rank card on the flop.

106 total combos removed 164 combos still in range. Bet 16.4 combos to maintain 10%.

Trips on board no A,
Unpaired hands effect: One Ax rank hand is reduced by 3/4 combos. 16/4 = 4×3 = 12. So 12 combos reduced from range.

Paired hands effect: 1 pair hand is completely eliminated from your range. So 6 less combos in your range.

Total combos removed 18. Total combos remainin in range 252. Bet 25.2 combos to maintain 10%.

Trips, A trips:
Unpaired combos effect: 3/4 of your unpaired hands are out of your range. 192/4 = 48×3 = 144. So 144 combos reduced from your range.

Paired combos effect: same, all 6 combos of AA are gone:

150 total combos gone, 120 remain so bet 12 combos to maintian 10%.

Now if you just want to pick a number of combinations to bet as a general guideline you can add these up and average them. It comes out to about 20 combos or so.

Should be noted that card removal from your opponents ranges will also effect your range in a similar way, but way too complex to factor that in.

Now lets look closer at this case:
3 different cards (no A) => a betting range of 24.9 combos to maintain 10%

for flop 2s 3s 4s we rougly have these (27) combos sorted by equity vs random:
2s 3s 4s - As 5s 1.00, As 6s 0.98, As 9s 0.98, As 7s 0.98, As Ks 0.98, As Ts 0.98, As Qs 0.97, As 8s 0.97, As Js 0.97, As 5c 0.91, As 5d 0.90, As 5h 0.90, As Ah 0.85, As Ac 0.85, As Ad 0.85, 5s Ac 0.81, 5s Ah 0.81, Ks Kc 0.81, Ks Kh 0.81, Ks Kd 0.80, 5s Ad 0.79, Qs Qh 0.78, Qs Qc 0.78, As 4d 0.78, Qs Qd 0.77, As 4h 0.77, As 4c 0.77,

for flop 2s 7h Kd we rougly have these (27) combos sorted by equity vs random:
2s 7h Kd - Ks Kc 0.99, Ks Kh 0.99, Kh Kc 0.98, 7s 7c 0.97, 7d 7c 0.97, 2d 2c 0.97, 7s 7d 0.97, 2h 2c 0.96, 2h 2d 0.96, As Ks 0.90, Kh Ac 0.90, As Ad 0.90, Ac Kc 0.90, Ks Ah 0.90, As Kc 0.90, Ah Kh 0.90, As Ah 0.90, As Kh 0.89, Kh Ad 0.89, Ah Kc 0.89, Ad Ac 0.89, Ah Ad 0.89, Ad Kc 0.89, Ks Ad 0.89, Ah Ac 0.88, As Ac 0.88, Ks Ac 0.88


Now lets look for both flops at the equity of combo No25 and No26 (No 25 is the last that we should include in the betting range to maintain 10%)

2s 3s 4s => No25: Qs Qd - 0.77, No26: As 4h 0.77
2s 7h Kd => No25: Ah Ac - 0.88, No26: As Ac 0.88

Looking at the equities we are better dumping 2s 3s 4s => No25 thus leaving this flop with 24 comboes to bet and instead include 2s 7h Kd => No26 in the betting range.
We still maintain 10% betting overall but with hands with better equities(vs random).

So how do we calculate these across all flops?

How did you get equity calcs for each of those hands without playing them vs another range?

Also look how silly your arbitrary 10% betting range is. You want to exclude hands from your betting range that have as much as 77% equity vs whatever range you're putting them against.

That does not seem to make sense to me.

To continue on your going to have to start grouping flops in some fashion. There maybe academic papers online that contain good grouping categories for playing algorithms at the Univesity of Alberta website (I think they're the ones someone else here may be able to confirm or deny that). But yea your going to start cayegorizing flops which may not be all that easy.

Just bet KK+ AKo and AKs. - There you go 10%

I just saw this - flop categorization is required.

A-x-x = bet 3 of a kind + 2pairs
A-6-6 (any paired ace board) bet AK and full houses
Three broadway flops not containing ace - bet straights and 3 kind

List continues......