# Equity with ranges

What equity do I have in the following situation:

Board: 3⋄9♥A♠7♠Q♥

Hero: K♠K♣

Villain 1: Top 25%

Villain 2: Top 25%

According to this calculator, I have 32.80%. But I don't believe that.

The range of 25% contains the following hands (correct me if I'm wrong)

{'3AS', 'QAS', '8AS', '6AS', 'TT', 'KAS', '5AS', '88', 'QKS', '66', 'TKS', '4AS', '6AO', '7AS', 'AA', '77', 'JAS', 'KAO', 'QQ', '9KS', 'TKO', 'JAO', '9AO', 'KK', 'QKO', 'TAO', 'QAO', '7AO', 'TJS', '5AO', 'JKO', 'JQS', '8KS', 'TQS', '8AO', 'JKS', '4AO', 'JJ', 'TAS', '9AS', '55', '99'}

Of these 42 hands 22 contain an ace, which would beat my hands. So only 48% do not contain an ace, this to the power of 2 would mean I have only a 23% probability that none of the others contain an ace. This alone should bring my equity below 23%. In total my own equity calculation gets it down to 11%.

Any suggestions why I deviate from the above link would be appreciated.

I'm using the equity calculator for my bot: www.github.com/dickreuter/poker. You are welcome to join the project, it's all open source if you have any good input.

• Can we see your personal equity calculation? – Grinch91 Nov 14 '16 at 11:29
• Sure. It's not the most efficient one but it works otherwise. github.com/dickreuter/Poker/blob/master/decisionmaker/…. When you run it it will do that for the above problem. To run it you need Python (best get anaconda, and pycharm). You can also download the full bot. – Nickpick Nov 14 '16 at 12:01
• I just wanted to note that just because 22 of the 42 hands listed have an ace does not mean that 52% of the range has an ace. That's because not every hand listed is equally likely. For example, there are 4 combinations of cards that give you A9s, but 12 combos that give you A9o. However, this ends up making aces even more likely, so my hunch would be that the 32.8% from the calculator is only using a single opponent hand rather than both. – Dr.DrfbagIII Nov 14 '16 at 14:45
• Check other equity calculator. I suggest you run a large simulation of just 5 cards and see if you get the proper probabilities. And I suggest you accept some answers. – paparazzo Nov 14 '16 at 14:58
• @Paparazzi Accept what answer? So far there is none. If you look at the code you see there are plenty of tests and it passes them all, it's just for ranges that I deviate and I'm not sure which one is correct now. Why simulate only 5 cards. Makes no sense. In holdem there are 7. My question is on the intuition of the above problem. I don't see how equity can possibly be 32%. – Nickpick Nov 14 '16 at 15:05

There are 169 possible starting hands, so the top 25% is the top 42(.25) hands. As per this page, the top 25% of all starting hands are therefore:

`AA,KK,QQ,AKs,JJ,AQs,KQs,AJs,KJs,TT,AKo,ATs,QJs,KTs,QTs,JTs,99,AQo,A9s,KQo,88,K9s,T9s,A8s,Q9s,J9s,AJo,A5s,77,A7s,KJo,A4s,A3s,A6s,QJo,66,K8s,T8s,A2s,98s,J8s,ATo`

I haven't compared this to your stated top 25% range, but it has the same number of hands in it, so I expect we agree on what the top 25% of hands are.

There are 6 ways to make each pair, 4 ways to make each suited hand and 12 ways to make each unsuited hand. Based on this we can establish the distribution of hands over the range and therefore that each pair makes up 2.48% of the range, each suited hand is 1.65% of the range and each unsuited hand is 4.96% of the range.

However, we need to adjust this for some hands which are no longer possible because of the known cards - for example there are only three `A7s` hands available because the A♠ and 7♠ are already dead on the board and there is only one `KQs` possible: K⋄Q⋄.

Next, we can evaluate each hand against our `KK` on this board to establish whether it wins or not. The hands which can beat `KK` on this board are `Ax`, `QQ`, `99`, `77`, `33`, `Q9`, `Q7`, `Q3`, `97`, `93`, `73`, so we attribute a value of 1 to these hands (that appears in our list of 42) and 0 to the rest (except `KK` which gets 0.5 for a tie), then multiply this value by the proportion of the range made up by these hands before summing for the entire range.

This gives us a final result of 49.0385% for the top 25% of hands versus our `KK` on this board. If we're roughly a flip against one top 25% hand, I'm not surprised that we're around 1/3rd against two hands from that range.

• I went ahead and compared this to ProPokerTools and interesting they give a (slightly) different number: 42.05%. The output from the tool shows an exhaustive run of 220 trials, where my method above only identifies 156 "ways" for the top 25% of hands to be made in this situation, so it looks like my combination cals must be slightly out, but hopefully this answer still demonstrates that it's not surprising we're flipping in this situation. – 3N1GM4 Dec 12 '16 at 15:37