Hey guys,
I got very interesting idea about making prop bets during regular NLHE cash game over at home games. Not sure if I created topic in proper section.

So here is the rules.
*THIS GAME IS RUNNING ON THE SIDE. LIKE A PROP BET*
We just imagine that I have 22 and another player AK we put equal amount of money and just see who got the best hand on the board -- WE DO NOT ACTUALLY GETTING DEALT ANY CARDS, we just imagine that. If the hand stopped on the flop we just ask dealer to finish the board since we got a prop bet on the side.
Here is the deal, they don't understand that they are drawing nearly dead if I make sets BUT I will hit my set more often since I can use 4 deuces that are in the deck instead of two like they are probably thinking.
So I made **** ton of money last night on this prop bet just because they are dumb and tend to gamble. I even gave them kind of discount of 5% cuz my edge is waaay higher than that. I actually put 5% on the top of my part.

So question is, what is the exact mathematical advantage does pair of deuces have in this case? How can I calculate it? I need it to get a margin of the discount I can make to keep it profitable in longrun.

We disregard cases when they win with 4card flush since we have different suites. Also he can win with straight when I make set, double paired board, boats over straights etc etc and vice versa. I'm not sure what is the equity in that cases.

I would really appreciate if you respond with formulas and mathematical explanation.

Thank you very much

What happens if the flop is 222 ? Do you automatically win?

OFC, who can beat five of the kind?
that's a good point btw

probably it will give them feeling that they got pwnt


also they can't beat quads with two cards to come, so it's a freewin automatically.

Ignoring straights, flushes, and boat>boat, 22 is about 59.4% to win. Since the things I ignored all benefit the AK, I'll estimate 57.5%

so 42.5% against 57.5%
so I consider 15% clear edge?

Hey man are u sure ur maths is right?
Im a friend of wnspi,and villain ask me for a discount of 10% of the bet if i win, for example we bet 100$ , i lose 100$, but lose 90$.
So if my math is right my ev here:
Ev=(90*0.575)-(100*0.425)=9.25$ for each bet right?
I also want to note that villain takes not AK but KQ offsuit.
Can you write a maths how u calc an equity?
How will changes our equity vs KQ?
How will changes our equity if they are will be suit?
How changes our equity if we will have not 22 but 99?
Thanks in advance, with great respect from friedrice88 and wnpsi

Yeah but I'll replace my estimate with a more exact figure, since I was purely guessing when I decided to subtract about 2% from 59.4% (I can't guarantee that 2% was enough to subtract). If it weren't for the 10% condition, I'd say you're without a doubt +EV, but with the condition I don't wanna promise that just yet.
KQ and 22/99 will barely change it, but suited will make a difference.

I have java code on another laptop that can simulate this without much tweaking. I'll probably feel more like simulating than doing the calculation, but either way I'll probably put this off to tomorrow.

Help me, bro, fishs are waiting!

Hey ! Can you post results???

Lol, amusing. Nice post OP

Can somebody count???

No, 42.5% vs 57.5% is a 35.29% edge.

I chuckle when I see people refer to 55% vs 45% as a "coin flip". It's a 22.2% edge. Sign me up all day every day.

EV = .55*1 - .45*1 = .55 - .45 = .10
Winning 0.1 on a $1 wager seems like a 10% edge, not a 22.2% edge. My definition of edge seems to match wikipedias (https://en.wikipedia.org/wiki/Gambli...antage_or_edge) which is to say EV/bet. (When the bet is 1, it's just the EV obviously)

I think you're doing
winprob/loseprob - 1
which is not any definition of edge I've ever seen.

Today i lost about 20 bets for about 100 hands, i bet 100$ vs 90$, but a also think this is small Ev
Note:
I bet on red 22
He bets on dark KQo

I think i have more than 55% here

That's fine, I was replying to this

I just read OP and have a question or two.

First, do you do this on every hand dealt or every hand that goes to a flop or what? Of course, NLHE boards are notoriously "biased" away from Aces and Kings since these are more likely to appear in players' hands when they play a hand. In terms of this exercise, are you ignoring this "bias" or do you want to take the "bias" into account?

Second, I can see why you may want to exclude hands that would have theoretically made a flush on a board with 4 of a suit since you presume that you both have two suits covered (say two red deuces vs. AsKc). But what about boards with 5 cards of a suit? That is rare, of course, but it definitely would be to the advantage of the AK player versus 22.

I see heehaww has said above that he will soon post results of a simulation but it looks to me that he hasn't popped back in.

If he doesn't post soon, I could write a program that can run through all the cases.

1)It is hard too understand 1st question, what is bias? So we make a deal before hand starts and agree with an amount of the bet.
2)we count to that winning AKo by flush is 2,5% and win 22 by a flush is only 2%
How long time it takes to write and make a simulation???

so 22 in the hand have 53% equity vs AKo(simulated on propokertools.com)
How u can say that we have 55-57% if all deuces in the deck ?
So calculator calcs 53% when we have 22 in the hand and only two deuces in the deck
but my prop bet is two deuces are virtual in the hand and we have 4 outs(4 deuces in the deck)
Also calculator calcs one A and one K and in the deck 6 outs
but in my prop bet is virtual AK and 8 outs
So I think I have rather better equity than 55% with 22

He's saying that if there is a flop it's likely the players that are actually playing the hand will have a disproportionately high amount of Ax and to a lesser extent Kx hands, making it LESS likely As or Ks show up on the board.

I think the most important question here is how does the chances of hitting minimum a set with 22 and a pair with KQ change in this kind of situation.
All the other cases like straights, flushes and so on are not that much effected by the bet and the chances will stay "about" the same i think and doesnt matter that much.

Under normal circumstances KQ hits minimum a pair in 49% of the time and 22 hits minimum a set 19% of the time.

In your circumstances (playing imagined hole cards) the numbers changes to:
KQ hits minimum a pair 58% of the time (9% mor than under normal circumstances which means a relative increase of 18%).
The chances of hitting minimum a set with 22 changes to 34% (15% more than before with an relative increase of 79%).

This advantage sounds huge because in 34% of all boards your opponent needs at least a straight to beat you.

KQ hits at least a straight only about 10% on the river under normal circumstances.

But finally i think the advantage and gain of equity in total is less than 10% and i would be surprised if you were more than 60% favorite in this type of bet. Reason why is that you "only" get huge favorite in 15% more of all boards and you have to discount that standard equity that you would already have for these 15% of boards. Also his chances raises in the 66% of cases where you don`t hit minimum your set.

You ask if you could give a discount from 100 to 90 for your opponent. For me this sounds still ok because you need only 52,6% equity for that. I think also you have (like stated before in this thread) about 57,5% equity. If it would be less than 55% i would be really surprised if more than 60% also.

I am not sure what is known at this point or not, so I will just amplify on Controlling's post above. It is straightforward math to derive the frequency of different boards (as Controlling did above).

OP asked about the formulas so here goes -- I imagine that everybody may already know these formulas but whatthehell.

In a standard deck of 52 cards, there are 4 2's, 8 A-K's and 40 Q-3's.

So the prob of getting at least one 2 on a 5-card board is most easily derived as being 100% - prob of getting NO 2's on a 5-card board. Which is 100% - [C(48,5)/C(52,5] = 34.1% where C(X,Y) is the Choose function or the number of combinations in which you can choose Y items out of X without replacement and when the order does not matter.

Similarly, it is easy to derive the prob of getting a board with no Aces, Kings, or Deuces as 100% - [C(40,5)/C(52,5)] = 25.3%.

Finally, prob of getting at least one Ace or King but NO deuces is = [C(8,1)*C(40,4) + C(8,2)*C(40,3) + C(8,3)*C(40,2) + C(8,4)*C(40,1) + C(8,5)*C(40,0)]/C(52,5) = 40.6%

Other probs along these lines can easily be derived as desired. Like prob of one deuce and two aces on board, etc.

So, simply taking these high-level probabilities and ignoring straights and flushes, we would imagine that 22 would win around 25.3% + 34.1% = 59.4% and that AK would win around 40.6% of the time.

As mentioned these are simple approximations of the true probs since they have ignored straights and flushes and complicated combos of both Aces/Kings and Deuces.

Again, I apologize if this information has already been derived, described, and discussed in the thread already.

Alright, since these are pretty easy to do, let's do a few more cases.

(1) AK makes broadway straight with QJT on board. Other two cards cannot be QQ, JJ, or TT which would give 22 a full house (we don't need to subtract full houses with another 2 on board or quad twos since we have previously assumed 22 would already win with any 2 on board) or AK which would be a chop (so let's subtract half of these since a chop is essentially a half-win).

4Q, 4J, and 4T in deck. Other two cards on board can be anything except QQ, JJ, or TT or half of the AK combos. So I get this prob as (4^3)*[C(49,2)-(3C(3,2)+(4^2)/2))]/C(52,5) = 2.85%.

(2) AK wins with a wheel straight with a 2345 on board. Fifth board card must be K-7 for AK to win (a Six or Ace would be a chop so let's count half of these). I get this prob as (4^4)*[32+((4+4)/2)]/C(52,5) = 0.35%.

(3) AK wins with five spades or clubs on board. The case of exactly four suited cards on board is a wash as they each have two suits covered. We need to exclude half of the rare cases of a straight flush on board since those would be chops. So I get this to be 2*[C(13,5)-(10/2)]/C(52,5) = 0.10%.

Putting this altogether with the previous post, it seems like AK would win around 40.57% + 2.85% + 0.35% + 0.10% = 43.87%. Let's round this and say AK would win around 44% and 22 would win around 56%.

Note that throughout we have been a little sloppy since many of the cases presented so far do have slight overlaps (or not all deals in the case are clear winners one way or the other). But as back-of-the-enveloping calculations go, these should get fairly close to the exact answer.

Comments welcome, especially if I screwed up somewhere.

great post,and excelent math master! thx!!!