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Hand Ranges, Combos, and EV
6 combos of pocket pairs — 13 types — 6 * 13 total combos
(AA, TT, 55)
16 combos of non-pairs — 78 types — 16 * 78 total combos
(AK, JT, 87, 23)
Starting Hand Combos
= (6 * 13) + (16 * 78) = 1326 combos
Since each pocket pair has 6 combos, each type represents 6 / 1326 = 0.45%.
Since each non-pair has 16 combos, each type represents 16 / 1326 = 1.2%.
Example 1 —
Now, let’s say you decide that to raise UTG with strictly TT+, AQ+, KQ
This range has pairs (TT, JJ, QQ, KK, AA) and non-pairs (AK, AQ, KQ).
This range represents (5 * 0.45%) + (3 * 1.2%) or about 6% of hands.
Example 2 —
If the button has been actively stealing the blinds, you want to fight-back from the SB with some frequency. You decide to do this with any pair and any broadway hand.
pairs (22, 33, 44, 55, 66, 77, 88, 99, TT, JJ, QQ, KK, AA)
non-pairs (AK, AQ, AJ, AT, KQ, KJ, KT, QT, QJ, JT)
This range represents (13 * 0.45%) + (10 * 1.2%) or about 18% of hands
Now, let’s use this idea to determine when to hold ’em and when to fold ’em.
Example 3 —
A tight opponent w/ 30 bb raises from the Button. You estimate that this opponent will open with any pair and any broadway or 18% of hands from this position. The flop is AXX. Now, this ace rag flop should diminish some of that range significantly. The opponent will likely fold everything except (AA, AK, AQ, AJ, AT). This means that a starting range of 18% will shrink to (1 * 0.45%) + (4 * 1.2%) or 5.25%. To keep the numbers simple, I’m going to say that the opponent continues with 6% and folds 12% of the 18% starting range. By betting on this ace high flop, the opponent will fold 2/3rds of their range!!!
Example 4 —
Same scenario but the flop is TXX. This 10 high flop is different. The opponent will continue with (TT, JJ, QQ, KK, AA) and (AT, KT, QT, JT) or ~7%. This means they opponent will call with 7% and fold 11% of the starting range 18%.
Example 5 —
Let’s add some bet sizes so we can calculate EV.
Our opponent raises to 3 bb and we call from the bb. The pot is 6.5 bb. If you decide to steal, you will be 4 bb into the 6.5 bb pot.
On an Ace high flop, (AA, AK, AQ, AJ, AT) will either call or re-raise if you try to steal. For the sake of argument, let’s say you win 10% of the time against this range.
All the other opponent hands will be assumed as a fold
— 18% total
— 5.25% calls or raises
— 12.75% folds
foldEV
= (%fold)(amount if fold)
= (12.75%/ 18%)(6.5 bb)
= 4.60 bb
callEV
= (%call) * [(%win)(amount win if call) – (%lose)(amount lose if call)]
= (5.25% / 18%) * [(10%)(6.5 bb + 4 bb) – (90%)(4 bb)]
= (5.25% / 18%) * [1.05 bb – 3.6 bb]
= -0.743 bb
totalEV
= 3.86 bb
Seems like a decent steal spot to me 🙂
Can it be optimized?
Example 6 —
What is the ideal bet size when F=30%, C=70%, P=6.5 bb, and W=10%?
0 < (F)(P) + (C)*[(W)*(P + B) – (1 – W)*(B)]
0 < FP + CWP + CWB – CB + CWB
0 < FP + CWP + 2CWB – CB
FP + CWP < B(2CW – C)
B > (FP + CWP) / (2CW – C)
B > (30%)(6.5 bb) + (70%)(10%)(6.5 bb)) / (2*(70%)(10%) – (70%))
B > (1.95 bb + 0.455 bb) / (0.14 – 0.7)
B < 4.29 bb
Example 7 —
What is the ideal win % when F=30%, C=70%, P=6.5 bb, and B = 4 bb?
0 < (F)(P) + (C)*[(W)*(P + B) – (1 – W)*(B)]
0 < FP + CWP + CWB – CB + CWB
0 < FP + CWP + 2CWB – CB
CB – FP < W(CP + 2CB)
W > (CB – FP) / (CP + 2CB)
W > ((70%)(4 bb) – (30%)(6.5 bb)) / ((70%)(6.5 bb) + 2*(70%)(4 bb))
W > (2.8 bb – 1.95 bb) / ( 4.55 bb + 5.6 bb)
W > 0.85 bb / 10.15 bb
W > 8.37%
Example 8 —
To detect trends,
If you don’t win very often, should you bet more or less in general?
If you want to bet smaller, do you need to win more often or less often?
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6 Replies
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How often your opponent folds?
Red = 5%, Green = 10%, Blue = 15%, Yellow = 20%
Left Graph
– When you bet B, each curve represents the minimum % win needed to reach 0 EV.
When the pot is 6.5 bb, you are looking to steal on specific flops by betting (say 3 bb). Now, the red line on the left graph shows that you need to win just over 20% of the time to have +EV when they fold 5%.
— If you bet smaller than 3 bb, you need to win less often.
— If your opponent folds a 3 bb bet more than 5%, you need to win less often.
Right Graph
– When your hand wins W when called, each curve represents the minimum bet size needed to reach 0 EV.
When the pot is 6.5 bb, you should be stealing when you have enough hand equity to still win when called. If you have 0 chance to win, the risk is large because you have no call EV, only fold EV. If you win only 20% of the time, then betting 2 bb should be +EV against an opponent who calls only 5% of the time.
— If the opponent folds more often, you can bet smaller.
— If your hand is stronger than usual, you should bet larger.
(hands with 2 outs should bet smaller, hands with 10 outs should bet larger)
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Nice Job. This kind of post takes a lot of work. I usually run out of edits before I find all of my mistakes. And if I have to do it somewhere else first, then somewhere else is where it stays. I appreciate your🍻 sharing. You are doing the work others are too lazy to do. 🏆
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Thank you,
I worked hard at it.
I’ve found one typo — I wrote “be” instead of “bet” but the context was obvious.
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I am so impressed with this post, @arw man you always bring the details to your amazing posts. Keep it up! Hope w get to meet up in person again sometime soon.
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Some real good info here, and I can only imagine the hours put in to get this out!
Thanks for the read 🙂
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