ofc strategy and math - Greenstein vs Deeb video

At around the 25:15 mark of the http://www.youtube.com/watch?v=2TtyyM8K-aE video, Shaun Deeb notes that Barry Greenstein makes a big mistake on 11th street by placing his Qh in front instead of in the middle:

Barry Greenstein







Shaun Deeb (Dealer)







Obviously Deeb is correct because there is only one more 3 left in the deck so he is much more likely to foul this way.

Does anyone know how to do the math for this? In other words, how much more likely is Greenstein to foul because of this mistake? No matter what, he needs one of his last 2 cards to be 6, 7, 8, 9, T or J to avoid fouling in the back. If the Qh is in front and his back card is 7, 8, 9 or T then any 2, 4 or 6 is a foul. If the Qh is in the middle then the case 3 is a foul.

Also, our http://www.flopturnriver.com/chinese-poker/ ofc game will soon let players display hands the same way I displayed the one above for strategy discussions.

If Barry puts the queen in the top row, then there is a 13.3% chance to foul on the middle (4 outs to the foul, a 6 won't be a foul since it will complete the bottom line instead of fouling the middle). That can be calculated like this:

30 (cards left in the deck) = 100%
4 (outs) = x %

x = 100*4/30 = 13.3%

And the chance to complete his bottom line equals to 50%.

30 = 100%
15 (outs) = x %

x = 100*15/30 = 50%

If, on the other hand, Barry puts the queen in the middle, then there is only a 3.3% chance to foul in the top row (1 out to a foul) and still a 50% chance to hit the bottom line, so yeah, seems that Barry increased his fouling equity here quite a bit by putting the Q on top

That video was really informative and certainly worth watching. I'm new to this game and I'm curious about whether or not there's a benefit to playing aggressively to "fantasy land" or conservatively over the long-haul. Even with two wildly different styles, both Deeb and Greenstein both fouled two hands in the training video. I know those hands represent only two examples of how given hands might be played, but I'm wondering if, over the course of thousands of hands, there's a predictable difference between the results aggressive players would experience compared to the wins/losses of their comparatively conservative counterparts. Any thoughts?

Originally Posted by Filik

If Barry puts the queen in the top row, then there is a 13.3% chance to foul on the middle (4 outs to the foul, a 6 won't be a foul since it will complete the bottom line instead of fouling the middle).

If 12th street is 7, 8, 9 or T then it goes in back and a 6 on 13th street is a foul (along with a 2 or 4 on 13th).

@dlbarlowe: I'm sure the game will be GTO solved at some point, but it's new enough to be still under investigation.

It's pretty easy to solve whether you have 50% equity to make a flush, given the board. However, that's a very specific question. If you're asking a general question like, "How should I play my 6th card, given the board?" then that's got a lot of variables.

Questions like, "How likely is it that I can make XXXXX hand, given the board?" are easy to answer.
Questions like, "How likely is it that I can make XXXXX hand or better in the middle, AND the bottom improves/remains good enough to avoid foul, given the board?" are a bit harder to answer, but still possible.

The huge hurdle is the sheer number of board combinations in the early and middle game combined with the huge number of run-outs. Late game becomes easier, since so much info is available. I suspect we'll see some optimal late game strategies before early game strategies. Once each hand is defined as a pairing hand, or straight/flush draw hand, you can eliminate some of the terms in the combinatoric sum and set more constricted boundaries on the possible hands for each tier.

Originally Posted by MadMojoMonkey

I suspect we'll see some optimal late game strategies before early game strategies. Once each hand is defined as a pairing hand, or straight/flush draw hand, you can eliminate some of the terms in the combinatoric sum and set more constricted boundaries on the possible hands for each tier.

yeah, def. however, so much of the hand depends on being set correctly. a mistake on the set or 5th - 7th card can really mess up your hand, so if you develop a good early game you should be able crush for a while.

I'm still not sure of the math but we can describe it like this:

Qh up front foul scenarios:
2, 4 or 6 with our 7, 8, 9 or T outs (2, 4 or 6 with 6 and J outs is ok)

Qh in the middle foul scenarios:
3h with any of our outs (6, 7, 8, 9, T, J)

Obviously we foul either way if we don't get one of our outs: 6, 7, 8, 9, T or J.

Originally Posted by Eric

If 12th street is 7, 8, 9 or T then it goes in back and a 6 on 13th street is a foul (along with a 2 or 4 on 13th).

I calculated the odds for one street only, I don't think it's worth calculating percentages for 2 cards to come since 12th street can bring a lot of cards that will change the percentages drastically. For example, if Barry gets an A, K, Q, 5, or 3 on 12th then his only play will be to put them in the middle and he'll be left with a 46.5% chance to miss his bottom line and foul (This percentage will change depending on what card Shaun hits on 12th). So yeah, there's pretty much always going to be a chance for him to foul on 13th regardless of whether he completes the bottom line on 12th or not, just that the chance of him fouling the middle if he does complete the bottom is far lower then if he missed the bottom on 12th

I also posted this on 2p2 but it is time to just answer it myself.

We need to know the probability of fouling under Scenario A vs Scenario B.

We are drawing two more cards and if at least one of the two cards is not 6, 7, 8, 9, T or J then we foul under both scenarios. We'll call these cards our outs.

Scenario A:
We also foul if we draw the last 3 in the deck (3h) with any of our outs.

Scenario B:
We also foul if we draw 2 or 4 with our 7, 8, 9 and T outs.
If we draw our 7, 8, 9 or T out before drawing the 6 then we also foul. This is the only spot where the order matters.

Cards Exposed: 22

Cards Left for Drawing Pool: 30

Breakdown of cards left for drawing:
2: 2
4: 2
3: 1 (only the 3 of hearts is left)
6: 2
7: 2
8: 2
9: 3
T: 2
J: 4 (all four of the jacks are live)
Other: 10
Total: 30

This is the probability for missing our outs altogether such that we foul under both scenarios:
P(blank and blank) = (15/30)*(14/29) = 24%

The order is important for part of Scenario B so we'll use permutations for the scenarios instead of combinations.

The number of ways we can draw our last 2 cards in our sample space:
Order Matters ==> Permutations ==> P(n,k) = n!/(n-k)! = P(30,2) = 30!/28! = 30*29 = 870

Scenario A: 3h issue. If the case 3 comes with any of our 15 outs in either of 2 orders (3h, 7c or 7c, 3h) then we foul. The chances are 15*2/P(30,2) or 3.5%.

*This means the total chances of fouling under Scenario A are 27.5% (24 + 3.5).

Scenario B: Deuces and Fours Issue. There are 4 deuces and fours left (two of each). If these combine with 9 of our outs (7, 8, 9, T, J) in either of 2 orders then we foul. The chances are 4*9*2/P(30,2) or 8.3%.

Scenario B: Sixes Issue. There are 2 sixes left. These 2 sixes cause fouls only if they come AFTER 9 of our outs (7, 8, 9, T). The chances are 2*9/P(30,2) or 2%.

*This means the total chances of fouling under Scenario B are 34.2% (24 + 8.3 + 2).

Let me know if these probabilities look correct. We'll need to do more of this type of thing if we are to build a foul calculator.



Notes to Self:

We could have used combinations instead of permutations for the parts where order doesn't matter:

Order Doesn't Matter ==> Combinations ==> C(n,k) = n!/k!(n-k)! = C(30,2) = 30!/(28!*2!) = (30*29)/2 = 435

3h Issue. If this case 3 comes with any of our 15 outs in any order then we foul. The chances are 15/C(30,2) or 3.5%

Deuce Four Issue. There are 4 deuces and fours left (two of each). If these combine with 9 of our outs (7, 8, 9, T) in any order then we foul. The chances are 4*9/C(30,2) or 8.3%

Funny that both of these guys play this game a ton, and even still make mistakes like this.

This game is so complex haha

Originally Posted by sam1FTR

Funny that both of these guys play this game a ton, and even still make mistakes like this.

This game is so complex haha

Yeah, I don't think Barry would make that type of mistake in a real game for money.