deal at final table, 3 way

final table, ive got 14000 to 12000 and 8000.


blinds 300,600.


1st 880

2nd 520

3rd 340


i chopped it 3 ways (580 each).


thats ok, right?


brad

That was a huge mistake, even if u dont know the exact figures you can easily see that you have the most chips and that you should get a bigger piece of the pie. Second pays 520 and you were happy to get 580 with the biggest stack i dont get this????

Not if you like money :-).


Why should the guy with 8K chips get as much as you ? No doubt people will tell you that "it's a crapshoot, anyone can win", but there is definitely skill involved playing short-handed even if blinds are very high, and to be honest, 300-600 is not all that high 3-handed with 34K chips on the table. Remember that, as chip leader, you cannot be busted in one hand. The middle stack can be busted by you and the short stack by either of you.


I calculate the true equity here to be $626 for you, $595 for the middle stack and $519 for the short stack. $45 doesn't sound like much maybe but if the three of you were sitting around the table and the short stack said "hey guys, please give me $45 and $15", what would you say ?


In the real world there are reasons why you might want to take a -EV deal like this - if your bankroll is very low or you feel your opponents are much better short-handed players than you. Still, if the second of those applies, better get some short-handed practice :-).


Finally, I am sure you are wondering where I got my numbers from. I have put together a computer applet which can do this calculation, and it is available for free on my website www.pokersoft.co.uk (http://www.pokersoft.co.uk), go to the Freeware page. Mason, if you think this is inappropriate, I would be happy to repost without this final paragraph.


Andy.

thanks. that was just what i wanted to know.


as to my reasoning, well, i play with the two guys a lot and i figured +- 50 bucks no biggee.


but im going to your site and prethink this for when im in a bigger tournament.


brad

Deals, like this should only be acceptable to you if you think that you are not capable of holding your own in th 3 way play. Early on, when my bankroll was less and my skill was even lower, I made a few of these types of deals to "get it over with" if I was tired or If I thought that I might lose a few hands and not get offerred the same deal in reverse.


At this point, I don't make any deals, regardless of how generous they are (unless a guy wants to just concede the tournament to me and i can only get a worse deal by playing it out, losing, and getting second. My skill level is such that I now feel I can win any final table regardless of where I stand so I play on. There was one exception in the last year: A guy who had less chips that I actually gave me $100 more than 1st because he didn't want to play it out. Yeah, there are some true idiots out there.


But in the end result you have to answer to yourself: are you happy with the deal? If you took it I sense that at least subconciously, you felt that you were above your head and wanted the title enough to give up the additional money you might have won. You need to do some soul-searching and ask yourself why you felt that this deal was adequate. If you thought it was right then who are we to question it... but you brought it up so I think you at least feel that you gave up something here.


Keep playing hard!

It's okay, but you might want to explain ho your algorithm does it's calculations.

Brad, this is not an insult. Your question would suggest that you're somewhat new to this. As such, it is the final table of a bigger tournament where you're probably going to be more correct to take a deal like this, because it's more likely that you're going to be up against two opponents with superior skills. In a small event, the skill level of the opposition is likely to be lower, so that's the time NOT to deal unless the deal is favorable to you.


I mean, if you get to 3-handed at the next WSOP main event, and turn down a deal for 1.2M because you think your chips are worth 1.3M, and you then bust out next and only get paid 0.7M, you are probably going to be kicking yourself for the rest of your life for missing out on that extra half a million.


If you say no to this deal in this small tourney and lose out on an extra couple of hundred, how long can you stay mad at yourself?


Later, Greg Raymer (FossilMan)

Mason,


Just like how you explain it in Gambling Theory and Other Topics. Now everyone's happy :-).


In fact this is really two questions - how do you calculate it theoretically and how do you estimate it at the table when you are trying to work out a deal. The theory is in Mason's book as I say. In practice, the starting point is that whatever percentage of the total chips you have, that is your percentage chance of winning, all other things being equal (which of course they aren't, never mind). Heads up is easy, say 1st place is $1000 and second $500, you have $500 locked up and are playing for another $500. If you have 60% of the chips your equity is 60% x $500 = $300, added to the $500 locked up makes $800 for you and $700 for your opponent.


Multi-way is more complicated. If you have 50% of the chips you still have a 50% chance of winning ; you then need to estimate your chances of finishing second if you don't win. For example, if the remaining chips are split equally between your opponents, it turns out your chances are 50% to win, 33% second, 17% third. Check the theory for the calculation (though I can expand here if necessary). If one of your opponents has one tiny chip, your chances are 50% first, almost 50% second, a very small % third. OK ?


In practice it is useful to remember some "benchmark" situations especially 3-handed. In the tournaments I play in (it depends on the payout structure) : if I have 20% of the chips I should expect around 2nd place prize money ; if I have 33% of the chips I expect 1/3 of the total money on offer ; and if I have 50% it's a little under the mid point between 1st and 2nd. Depends on the distribution of the remaining chips but depends a lot more on your opponents' ability so there you go.


Now, of course there are other ways that the money can be split. One you might hear is to split all the remaining money percentage-wise according to chip count. So if the prizes are $2000, $1000 and $500 then take $500 each and multiply the remaining $2000 by the percentage of the chips each player has. This favours the large stacks (you can easily construct a scenario where the large stack is "worth" more than first prize using this method). Nix it if you have a small stack, nod, smile and go "yes that sounds fair" if you have a large one. Even suggest it with a large stack if you think you might get away with it.


More common over here is for someone to say something like "OK, let's make 2nd and 3rd both $1000 and 1st $1500". Natch, this favours the small stack at the expense of the big one.


Just take your time and remember you can always get exactly what your stack is worth by politely saying "No". Don't let anyone push you into a deal you don't want, and if they start getting personal, ignore them or take appropriate action depending on who is harder :-).


Andy.

well, i only missed out on say 50 dollars.


and by bigger tournaments i mean the nighttime tournaments where in the exact same situation against the exact same opponents i would insist on more money or just play, because then we would be talking 500 - 1000 difference.


but i am pretty much a total novice at tournaments.


brad

well, i just wondered how much i gave up (i didnt figure it was a lot). one hand before we chopped i had the 8000 stack and said lets just chop, its pretty close (when you figure on going to a live game).


then i won a big hand and had the chip lead, they said ok lets chop, so i said ok.


brad

I cannot run .exe files since I don't have that type of computer. However I am curious about the algorithm. Do you have an explicit formula, based on stack sizes, or do you some kind of simulation to get an estimate? I read `Gambling Theory ..' (a while ago) but I remember feeling unsatisfied (no offense) by the treatment of tournament deals. I will have to look at it again to see exactly what it says. For two-way it is clear, and for three-way it seemed that the probability of coming third is inversely proportional to stack size, so you can get an explicit formula. But I couldn't see what to do for N players.


Is there a known explicit formula? I could maybe figure one out, but I don't want to bother trying if it's been done.


Dirk

Mason's "algorithm" is a heuristic but does not give an exact correct solution. It turns out there is a universal limiting solution for fair tournaments where the stakes are "small" relative to the bankrolls. This solution satisfies a partial differential equation. This is not unexpected, since the normal distribution and the two-player tournament solution satisfy the same partial differential equation with different boundary conditions. I wrote code to solve the three-player tournament and produced a table of probabilities of first, second, and last place depending on stack size. Unfortunately Mason lost it.

Dirk,


Essentially you are correct, for more than 4 players it is easier to run a Monte Carlo simulation which is what the program does. For 3 or 4 players, the program works out the probability of each combination (6 combs for 3 players, 24 combs for 4 players) using a simple formula which is effectively the same as that used by Mason in the book.


Kim as well, I hear what you are saying, but let's be pragmatic, this is only an approximation because in real life all players are not of equal ability. I'm confident that the numbers the app produces are close enough for a "starting point" when determining a fair deal in real life.


Andy.

That's what I thought. It seems you could simulate the situation for N players by a random walk in an N vertex simplex, except that once you hit a boundary (a player busts) you stay there, and do the walk in that smaller simplex, and contiue until you hit a vertex. The PDE's you mention would come from the limiting case as step size goes to zero.


I can see how this all goes, but I couldn't be bothered figuring it all out, especially since it has surely been done. So is there an explicit formula in terms of stack size (ignoring other factors)?


Dirk

I agree for practical purposes, the simulation is fine, especially for people who want to run some cases to get a feel for what is right when they have to make a decision at the table in a potential deal making situation. But I am interested in the theory.


How do you figure these probabilities. (And if you do it for four, why not 9. 9! is only 362880.)


Are you assuming (for the exact calculation) that if a player is eliminated (with some probability for that elimination that you calculate), then his chips are evenly distributed amongst the remaining players? I am not convinced that this gives the right answer. (Not at all.)


Can anyone give an authoratative answer on this?


Dirk

"And if you do it for four, why not 9"


Because I couldn't be bothered to write a properly recursive algorithm :-)


Andy.

Well, this means you got the worst of it both times, so to speak (you really didn't get the worst of it the first time, since there wasn't a deal, but you know what I mean).


These guys knew an even split was a bad deal, and they wouldn't do it when they had the lead, but would with you in the lead. Look and learn.


Later, Greg Raymer (FossilMan)

Yes, it is the expectation of the payoff from a Brownian motion (random walk) on a simplex. There is probably no simple formula, but I had solved it numerically with a finite-difference scheme.

Were your results different to those in Mason's book (other than numerical errors from discretization)?


Actually you could look at the formula implied by the book and see if it satisfies the pde's. Do you know if it does or not?


Also the appropriate simplex would have different topology from what one might expect. For example, with N=3 players, the vertices of the triangle would be duplcated, so each edge has its own two endpoints, disjoint from the other edges' endpoint --- a total of 6 vertices corresponding to the 6 rankings. I'm not sure how you would model this continuously, but I can see the discrete process.


Dirk

There are two different formulas in Mason's book for three-player payouts. (The section is written by Mark Weitzman (spelling?).) They do not satisfy the pde's so they cannot be right. The first formula assumes that if a players busts, his chips are evenly distributed amongst the other players. The second assumes that if a certain player wins, the others come second with probability proportional to initial stack size. Both are reasonable approximating assumptions, but they do not give the right answer.


Dirk

I just did this same simulation. The solution is the Dirchlet problem on an equilateral triangle.


More intuitively, consider a tournament with N chips in play but 3 players left. At each iteration, randomly one of the three players tosses a chip to the other player. This is the best model for a 3-player tournament without becoming game-specific.


Hence, you can represent the current chip count as a coordinate in an equilateral triangle, where each vertex represents a player having all of the chips and each side represents the situation where the player who is represented by the opposite vertex is out of chips. So you get a triangular lattice such that you take a walk in one of 6 random directions one unit parallel to one of the edges. Hence the equity at any point must be the average of it's 6 neighbors.


We know the equity solution on the edge. In fact we know P(first) is just proportional to chip count, so the equity question is reduced to P(third). We know the boundary conditions--P(third) is 1 on one edge and zero on the other two edges, hence there exists a unique solution in the interior of the triangle. Blow I have listed the 10 and 30 chip solutions in comparison with both malmuth formulas, and we can see both Malmuth formulas underestimate P(third) for small stacks, thus overestimating the equity of short stacks.


Here are the tables for P(third) for player A:


Chips 1 1 8, P is 0.495439, mal1 is 0.470588, mal2 is 0.488889

Chips 1 2 7, P is 0.667328, mal1 is 0.608696, mal2 is 0.641667

Chips 1 3 6, P is 0.739688, mal1 is 0.666667, mal2 is 0.707143

Chips 1 4 5, P is 0.768104, mal1 is 0.689655, mal2 is 0.733333

Chips 1 5 4, P is 0.768104, mal1 is 0.689655, mal2 is 0.733333

Chips 1 6 3, P is 0.739688, mal1 is 0.666667, mal2 is 0.707143

Chips 1 7 2, P is 0.667328, mal1 is 0.608696, mal2 is 0.641667

Chips 1 8 1, P is 0.495439, mal1 is 0.470588, mal2 is 0.488889

Chips 2 1 7, P is 0.305306, mal1 is 0.304348, mal2 is 0.311111

Chips 2 2 6, P is 0.463537, mal1 is 0.428571, mal2 is 0.450000

Chips 2 3 5, P is 0.539159, mal1 is 0.483871, mal2 is 0.514286

Chips 2 4 4, P is 0.561670, mal1 is 0.500000, mal2 is 0.533333

Chips 2 5 3, P is 0.539159, mal1 is 0.483871, mal2 is 0.514286

Chips 2 6 2, P is 0.463536, mal1 is 0.428571, mal2 is 0.450000

Chips 2 7 1, P is 0.305305, mal1 is 0.304348, mal2 is 0.311111

Chips 3 1 6, P is 0.205528, mal1 is 0.222222, mal2 is 0.216667

Chips 3 2 5, P is 0.324209, mal1 is 0.322581, mal2 is 0.325000

Chips 3 3 4, P is 0.377745, mal1 is 0.363636, mal2 is 0.371429

Chips 3 4 3, P is 0.377745, mal1 is 0.363636, mal2 is 0.371429

Chips 3 5 2, P is 0.324209, mal1 is 0.322581, mal2 is 0.325000

Chips 3 6 1, P is 0.205527, mal1 is 0.222222, mal2 is 0.216667

Chips 4 1 5, P is 0.140115, mal1 is 0.172414, mal2 is 0.155556

Chips 4 2 4, P is 0.219168, mal1 is 0.250000, mal2 is 0.233333

Chips 4 3 3, P is 0.244517, mal1 is 0.272727, mal2 is 0.257143

Chips 4 4 2, P is 0.219168, mal1 is 0.250000, mal2 is 0.233333

Chips 4 5 1, P is 0.140115, mal1 is 0.172414, mal2 is 0.155556

Chips 5 1 4, P is 0.091785, mal1 is 0.137931, mal2 is 0.111111

Chips 5 2 3, P is 0.136638, mal1 is 0.193548, mal2 is 0.160714

Chips 5 3 2, P is 0.136638, mal1 is 0.193548, mal2 is 0.160714

Chips 5 4 1, P is 0.091785, mal1 is 0.137931, mal2 is 0.111111

Chips 6 1 3, P is 0.054787, mal1 is 0.111111, mal2 is 0.076190

Chips 6 2 2, P is 0.072931, mal1 is 0.142857, mal2 is 0.100000

Chips 6 3 1, P is 0.054787, mal1 is 0.111111, mal2 is 0.076190

Chips 7 1 2, P is 0.027368, mal1 is 0.086957, mal2 is 0.047222

Chips 7 2 1, P is 0.027368, mal1 is 0.086957, mal2 is 0.047222

Chips 8 1 1, P is 0.009123, mal1 is 0.058824, mal2 is 0.022222


Chips 1 1 28, P is 0.499831, mal1 is 0.491228, mal2 is 0.498851

Chips 1 2 27, P is 0.680667, mal1 is 0.650602, mal2 is 0.664286

Chips 1 3 26, P is 0.767201, mal1 is 0.728972, mal2 is 0.746296

Chips 1 4 25, P is 0.816691, mal1 is 0.775194, mal2 is 0.794872

Chips 1 5 24, P is 0.848289, mal1 is 0.805369, mal2 is 0.826667

Chips 1 6 23, P is 0.869919, mal1 is 0.826347, mal2 is 0.848810

Chips 1 7 22, P is 0.885404, mal1 is 0.841530, mal2 is 0.864855

Chips 1 8 21, P is 0.896801, mal1 is 0.852792, mal2 is 0.876768

Chips 1 9 20, P is 0.905305, mal1 is 0.861244, mal2 is 0.885714

Chips 1 10 19, P is 0.911653, mal1 is 0.867580, mal2 is 0.892424

Chips 1 11 18, P is 0.916314, mal1 is 0.872247, mal2 is 0.897368

Chips 1 12 17, P is 0.919592, mal1 is 0.875536, mal2 is 0.900855

Chips 1 13 16, P is 0.921682, mal1 is 0.877637, mal2 is 0.903081

Chips 1 14 15, P is 0.922700, mal1 is 0.878661, mal2 is 0.904167

Chips 1 15 14, P is 0.922700, mal1 is 0.878661, mal2 is 0.904167

Chips 1 16 13, P is 0.921682, mal1 is 0.877637, mal2 is 0.903081

Chips 1 17 12, P is 0.919592, mal1 is 0.875536, mal2 is 0.900855

Chips 1 18 11, P is 0.916313, mal1 is 0.872247, mal2 is 0.897368

Chips 1 19 10, P is 0.911652, mal1 is 0.867580, mal2 is 0.892424

Chips 1 20 9, P is 0.905304, mal1 is 0.861244, mal2 is 0.885714

Chips 1 21 8, P is 0.896800, mal1 is 0.852792, mal2 is 0.876768

Chips 1 22 7, P is 0.885404, mal1 is 0.841530, mal2 is 0.864855

Chips 1 23 6, P is 0.869918, mal1 is 0.826347, mal2 is 0.848810

Chips 1 24 5, P is 0.848288, mal1 is 0.805369, mal2 is 0.826667

Chips 1 25 4, P is 0.816691, mal1 is 0.775194, mal2 is 0.794872

Chips 1 26 3, P is 0.767201, mal1 is 0.728972, mal2 is 0.746296

Chips 1 27 2, P is 0.680667, mal1 is 0.650602, mal2 is 0.664286

Chips 1 28 1, P is 0.499831, mal1 is 0.491228, mal2 is 0.498851

Chips 2 1 27, P is 0.318320, mal1 is 0.325301, mal2 is 0.331034

Chips 2 2 26, P is 0.498650, mal1 is 0.481481, mal2 is 0.495238

Chips 2 3 25, P is 0.607198, mal1 is 0.572519, mal2 is 0.592593

Chips 2 4 24, P is 0.677459, mal1 is 0.631579, mal2 is 0.656410

Chips 2 5 23, P is 0.725663, mal1 is 0.672515, mal2 is 0.700952

Chips 2 6 22, P is 0.760157, mal1 is 0.702128, mal2 is 0.733333

Chips 2 7 21, P is 0.785548, mal1 is 0.724138, mal2 is 0.757488

Chips 2 8 20, P is 0.804546, mal1 is 0.740741, mal2 is 0.775758

Chips 2 9 19, P is 0.818829, mal1 is 0.753304, mal2 is 0.789610

Chips 2 10 18, P is 0.829469, mal1 is 0.762712, mal2 is 0.800000

Chips 2 11 17, P is 0.837168, mal1 is 0.769547, mal2 is 0.807557

Chips 2 12 16, P is 0.842387, mal1 is 0.774194, mal2 is 0.812698

Chips 2 13 15, P is 0.845412, mal1 is 0.776892, mal2 is 0.815686

Chips 2 14 14, P is 0.846404, mal1 is 0.777778, mal2 is 0.816667

Chips 2 15 13, P is 0.845412, mal1 is 0.776892, mal2 is 0.815686

Chips 2 16 12, P is 0.842386, mal1 is 0.774194, mal2 is 0.812698

Chips 2 17 11, P is 0.837168, mal1 is 0.769547, mal2 is 0.807557

Chips 2 18 10, P is 0.829468, mal1 is 0.762712, mal2 is 0.800000

Chips 2 19 9, P is 0.818828, mal1 is 0.753304, mal2 is 0.789610

Chips 2 20 8, P is 0.804546, mal1 is 0.740741, mal2 is 0.775758

Chips 2 21 7, P is 0.785547, mal1 is 0.724138, mal2 is 0.757488

Chips 2 22 6, P is 0.760156, mal1 is 0.702128, mal2 is 0.733333

Chips 2 23 5, P is 0.725663, mal1 is 0.672515, mal2 is 0.700952

Chips 2 24 4, P is 0.677458, mal1 is 0.631579, mal2 is 0.656410

Chips 2 25 3, P is 0.607197, mal1 is 0.572519, mal2 is 0.592593

Chips 2 26 2, P is 0.498650, mal1 is 0.481481, mal2 is 0.495238

Chips 2 27 1, P is 0.318320, mal1 is 0.325301, mal2 is 0.331034

Chips 3 1 26, P is 0.230774, mal1 is 0.242991, mal2 is 0.246552

Chips 3 2 25, P is 0.387740, mal1 is 0.381679, mal2 is 0.392857

Chips 3 3 24, P is 0.495444, mal1 is 0.470588, mal2 is 0.488889

Chips 3 4 23, P is 0.571468, mal1 is 0.531792, mal2 is 0.556044

Chips 3 5 22, P is 0.626688, mal1 is 0.575916, mal2 is 0.605000

Chips 3 6 21, P is 0.667717, mal1 is 0.608696, mal2 is 0.641667

Chips 3 7 20, P is 0.698660, mal1 is 0.633484, mal2 is 0.669565

Chips 3 8 19, P is 0.722136, mal1 is 0.652361, mal2 is 0.690909

Chips 3 9 18, P is 0.739863, mal1 is 0.666667, mal2 is 0.707143

Chips 3 10 17, P is 0.752984, mal1 is 0.677291, mal2 is 0.719231

Chips 3 11 16, P is 0.762264, mal1 is 0.684825, mal2 is 0.727820

Chips 3 12 15, P is 0.768201, mal1 is 0.689655, mal2 is 0.733333

Chips 3 13 14, P is 0.771099, mal1 is 0.692015, mal2 is 0.736029

Chips 3 14 13, P is 0.771099, mal1 is 0.692015, mal2 is 0.736029

Chips 3 15 12, P is 0.768201, mal1 is 0.689655, mal2 is 0.733333

Chips 3 16 11, P is 0.762263, mal1 is 0.684825, mal2 is 0.727820

Chips 3 17 10, P is 0.752983, mal1 is 0.677291, mal2 is 0.719231

Chips 3 18 9, P is 0.739862, mal1 is 0.666667, mal2 is 0.707143

Chips 3 19 8, P is 0.722135, mal1 is 0.652361, mal2 is 0.690909

Chips 3 20 7, P is 0.698658, mal1 is 0.633484, mal2 is 0.669565

Chips 3 21 6, P is 0.667716, mal1 is 0.608696, mal2 is 0.641667

Chips 3 22 5, P is 0.626687, mal1 is 0.575916, mal2 is 0.605000

Chips 3 23 4, P is 0.571468, mal1 is 0.531792, mal2 is 0.556044

Chips 3 24 3, P is 0.495443, mal1 is 0.470588, mal2 is 0.488889

Chips 3 25 2, P is 0.387740, mal1 is 0.381679, mal2 is 0.392857

Chips 3 26 1, P is 0.230774, mal1 is 0.242991, mal2 is 0.246552

Chips 4 1 25, P is 0.179934, mal1 is 0.193798, mal2 is 0.195402

Chips 4 2 24, P is 0.314441, mal1 is 0.315789, mal2 is 0.323810

Chips 4 3 23, P is 0.414356, mal1 is 0.398844, mal2 is 0.413757

Chips 4 4 22, P is 0.489200, mal1 is 0.458333, mal2 is 0.479487

Chips 4 5 21, P is 0.545923, mal1 is 0.502392, mal2 is 0.528889

Chips 4 6 20, P is 0.589327, mal1 is 0.535714, mal2 is 0.566667

Chips 4 7 19, P is 0.622683, mal1 is 0.561181, mal2 is 0.595784

Chips 4 8 18, P is 0.648235, mal1 is 0.580645, mal2 is 0.618182

Chips 4 9 17, P is 0.667524, mal1 is 0.595331, mal2 is 0.635165

Chips 4 10 16, P is 0.681615, mal1 is 0.606061, mal2 is 0.647619

Chips 4 11 15, P is 0.691225, mal1 is 0.613383, mal2 is 0.656140

Chips 4 12 14, P is 0.696819, mal1 is 0.617647, mal2 is 0.661111

Chips 4 13 13, P is 0.698656, mal1 is 0.619048, mal2 is 0.662745

Chips 4 14 12, P is 0.696819, mal1 is 0.617647, mal2 is 0.661111

Chips 4 15 11, P is 0.691224, mal1 is 0.613383, mal2 is 0.656140

Chips 4 16 10, P is 0.681614, mal1 is 0.606061, mal2 is 0.647619

Chips 4 17 9, P is 0.667523, mal1 is 0.595331, mal2 is 0.635165

Chips 4 18 8, P is 0.648233, mal1 is 0.580645, mal2 is 0.618182

Chips 4 19 7, P is 0.622682, mal1 is 0.561181, mal2 is 0.595784

Chips 4 20 6, P is 0.589325, mal1 is 0.535714, mal2 is 0.566667

Chips 4 21 5, P is 0.545922, mal1 is 0.502392, mal2 is 0.528889

Chips 4 22 4, P is 0.489198, mal1 is 0.458333, mal2 is 0.479487

Chips 4 23 3, P is 0.414355, mal1 is 0.398844, mal2 is 0.413757

Chips 4 24 2, P is 0.314440, mal1 is 0.315789, mal2 is 0.323810

Chips 4 25 1, P is 0.179934, mal1 is 0.193798, mal2 is 0.195402

Chips 5 1 24, P is 0.146648, mal1 is 0.161074, mal2 is 0.160920

Chips 5 2 23, P is 0.262523, mal1 is 0.269006, mal2 is 0.273810

Chips 5 3 22, P is 0.353060, mal1 is 0.345550, mal2 is 0.356481

Chips 5 4 21, P is 0.423701, mal1 is 0.401914, mal2 is 0.418803

Chips 5 5 20, P is 0.478907, mal1 is 0.444444, mal2 is 0.466667

Chips 5 6 19, P is 0.522068, mal1 is 0.476987, mal2 is 0.503788

Chips 5 7 18, P is 0.555673, mal1 is 0.501992, mal2 is 0.532609

Chips 5 8 17, P is 0.581526, mal1 is 0.521073, mal2 is 0.554779

Chips 5 9 16, P is 0.600921, mal1 is 0.535316, mal2 is 0.571429

Chips 5 10 15, P is 0.614768, mal1 is 0.545455, mal2 is 0.583333

Chips 5 11 14, P is 0.623683, mal1 is 0.551971, mal2 is 0.591009

Chips 5 12 13, P is 0.628049, mal1 is 0.555160, mal2 is 0.594771

Chips 5 13 12, P is 0.628048, mal1 is 0.555160, mal2 is 0.594771

Chips 5 14 11, P is 0.623682, mal1 is 0.551971, mal2 is 0.591009

Chips 5 15 10, P is 0.614767, mal1 is 0.545455, mal2 is 0.583333

Chips 5 16 9, P is 0.600920, mal1 is 0.535316, mal2 is 0.571429

Chips 5 17 8, P is 0.581525, mal1 is 0.521073, mal2 is 0.554779

Chips 5 18 7, P is 0.555671, mal1 is 0.501992, mal2 is 0.532609

Chips 5 19 6, P is 0.522066, mal1 is 0.476987, mal2 is 0.503788

Chips 5 20 5, P is 0.478905, mal1 is 0.444444, mal2 is 0.466667

Chips 5 21 4, P is 0.423699, mal1 is 0.401914, mal2 is 0.418803

Chips 5 22 3, P is 0.353059, mal1 is 0.345550, mal2 is 0.356481

Chips 5 23 2, P is 0.262522, mal1 is 0.269006, mal2 is 0.273810

Chips 5 24 1, P is 0.146648, mal1 is 0.161074, mal2 is 0.160920

Chips 6 1 23, P is 0.122992, mal1 is 0.137725, mal2 is 0.135961

Chips 6 2 22, P is 0.223640, mal1 is 0.234043, mal2 is 0.235714

Chips 6 3 21, P is 0.304941, mal1 is 0.304348, mal2 is 0.311111

Chips 6 4 20, P is 0.370171, mal1 is 0.357143, mal2 is 0.369231

Chips 6 5 19, P is 0.422251, mal1 is 0.397490, mal2 is 0.414545

Chips 6 6 18, P is 0.463563, mal1 is 0.428571, mal2 is 0.450000

Chips 6 7 17, P is 0.495960, mal1 is 0.452471, mal2 is 0.477592

Chips 6 8 16, P is 0.520844, mal1 is 0.470588, mal2 is 0.498701

Chips 6 9 15, P is 0.539249, mal1 is 0.483871, mal2 is 0.514286

Chips 6 10 14, P is 0.551911, mal1 is 0.492958, mal2 is 0.525000

Chips 6 11 13, P is 0.559322, mal1 is 0.498258, mal2 is 0.531269

Chips 6 12 12, P is 0.561761, mal1 is 0.500000, mal2 is 0.533333

Chips 6 13 11, P is 0.559321, mal1 is 0.498258, mal2 is 0.531269

Chips 6 14 10, P is 0.551910, mal1 is 0.492958, mal2 is 0.525000

Chips 6 15 9, P is 0.539247, mal1 is 0.483871, mal2 is 0.514286

Chips 6 16 8, P is 0.520842, mal1 is 0.470588, mal2 is 0.498701

Chips 6 17 7, P is 0.495958, mal1 is 0.452471, mal2 is 0.477592

Chips 6 18 6, P is 0.463561, mal1 is 0.428571, mal2 is 0.450000

Chips 6 19 5, P is 0.422249, mal1 is 0.397490, mal2 is 0.414545

Chips 6 20 4, P is 0.370169, mal1 is 0.357143, mal2 is 0.369231

Chips 6 21 3, P is 0.304940, mal1 is 0.304348, mal2 is 0.311111

Chips 6 22 2, P is 0.223639, mal1 is 0.234043, mal2 is 0.235714

Chips 6 23 1, P is 0.122992, mal1 is 0.137725, mal2 is 0.135961

Chips 7 1 22, P is 0.105143, mal1 is 0.120219, mal2 is 0.116954

Chips 7 2 21, P is 0.193182, mal1 is 0.206897, mal2 is 0.205556

Chips 7 3 20, P is 0.265891, mal1 is 0.271493, mal2 is 0.274074

Chips 7 4 19, P is 0.325334, mal1 is 0.320675, mal2 is 0.327739

Chips 7 5 18, P is 0.373464, mal1 is 0.358566, mal2 is 0.370000

Chips 7 6 17, P is 0.411961, mal1 is 0.387833, mal2 is 0.403205

Chips 7 7 16, P is 0.442190, mal1 is 0.410256, mal2 is 0.428986

Chips 7 8 15, P is 0.465214, mal1 is 0.427046, mal2 is 0.448485

Chips 7 9 14, P is 0.481832, mal1 is 0.439024, mal2 is 0.462500

Chips 7 10 13, P is 0.492612, mal1 is 0.446735, mal2 is 0.471569

Chips 7 11 12, P is 0.497913, mal1 is 0.450512, mal2 is 0.476023

Chips 7 12 11, P is 0.497913, mal1 is 0.450512, mal2 is 0.476023

Chips 7 13 10, P is 0.492611, mal1 is 0.446735, mal2 is 0.471569

Chips 7 14 9, P is 0.481831, mal1 is 0.439024, mal2 is 0.462500

Chips 7 15 8, P is 0.465212, mal1 is 0.427046, mal2 is 0.448485

Chips 7 16 7, P is 0.442187, mal1 is 0.410256, mal2 is 0.428986

Chips 7 17 6, P is 0.411959, mal1 is 0.387833, mal2 is 0.403205

Chips 7 18 5, P is 0.373462, mal1 is 0.358566, mal2 is 0.370000

Chips 7 19 4, P is 0.325332, mal1 is 0.320675, mal2 is 0.327739

Chips 7 20 3, P is 0.265890, mal1 is 0.271493, mal2 is 0.274074

Chips 7 21 2, P is 0.193181, mal1 is 0.206897, mal2 is 0.205556

Chips 7 22 1, P is 0.105142, mal1 is 0.120219, mal2 is 0.116954

Chips 8 1 21, P is 0.091041, mal1 is 0.106599, mal2 is 0.101916

Chips 8 2 20, P is 0.168434, mal1 is 0.185185, mal2 is 0.180952

Chips 8 3 19, P is 0.233284, mal1 is 0.244635, mal2 is 0.243098

Chips 8 4 18, P is 0.286940, mal1 is 0.290323, mal2 is 0.292308

Chips 8 5 17, P is 0.330734, mal1 is 0.325670, mal2 is 0.331282

Chips 8 6 16, P is 0.365855, mal1 is 0.352941, mal2 is 0.361905

Chips 8 7 15, P is 0.393301, mal1 is 0.373665, mal2 is 0.385507

Chips 8 8 14, P is 0.413864, mal1 is 0.388889, mal2 is 0.403030

Chips 8 9 13, P is 0.428142, mal1 is 0.399317, mal2 is 0.415126

Chips 8 10 12, P is 0.436546, mal1 is 0.405405, mal2 is 0.422222

Chips 8 11 11, P is 0.439321, mal1 is 0.407407, mal2 is 0.424561

Chips 8 12 10, P is 0.436546, mal1 is 0.405405, mal2 is 0.422222

Chips 8 13 9, P is 0.428140, mal1 is 0.399317, mal2 is 0.415126

Chips 8 14 8, P is 0.413863, mal1 is 0.388889, mal2 is 0.403030

Chips 8 15 7, P is 0.393299, mal1 is 0.373665, mal2 is 0.385507

Chips 8 16 6, P is 0.365853, mal1 is 0.352941, mal2 is 0.361905

Chips 8 17 5, P is 0.330732, mal1 is 0.325670, mal2 is 0.331282

Chips 8 18 4, P is 0.286938, mal1 is 0.290323, mal2 is 0.292308

Chips 8 19 3, P is 0.233283, mal1 is 0.244635, mal2 is 0.243098

Chips 8 20 2, P is 0.168432, mal1 is 0.185185, mal2 is 0.180952

Chips 8 21 1, P is 0.091041, mal1 is 0.106599, mal2 is 0.101916

Chips 9 1 20, P is 0.079489, mal1 is 0.095694, mal2 is 0.089655

Chips 9 2 19, P is 0.147714, mal1 is 0.167401, mal2 is 0.160390

Chips 9 3 18, P is 0.205393, mal1 is 0.222222, mal2 is 0.216667

Chips 9 4 17, P is 0.253431, mal1 is 0.264591, mal2 is 0.261538

Chips 9 5 16, P is 0.292749, mal1 is 0.297398, mal2 is 0.297143

Chips 9 6 15, P is 0.324194, mal1 is 0.322581, mal2 is 0.325000

Chips 9 7 14, P is 0.348487, mal1 is 0.341463, mal2 is 0.346196

Chips 9 8 13, P is 0.366208, mal1 is 0.354949, mal2 is 0.361497

Chips 9 9 12, P is 0.377785, mal1 is 0.363636, mal2 is 0.371429

Chips 9 10 11, P is 0.383503, mal1 is 0.367893, mal2 is 0.376316

Chips 9 11 10, P is 0.383503, mal1 is 0.367893, mal2 is 0.376316

Chips 9 12 9, P is 0.377784, mal1 is 0.363636, mal2 is 0.371429

Chips 9 13 8, P is 0.366206, mal1 is 0.354949, mal2 is 0.361497

Chips 9 14 7, P is 0.348485, mal1 is 0.341463, mal2 is 0.346196

Chips 9 15 6, P is 0.324192, mal1 is 0.322581, mal2 is 0.325000

Chips 9 16 5, P is 0.292747, mal1 is 0.297398, mal2 is 0.297143

Chips 9 17 4, P is 0.253429, mal1 is 0.264591, mal2 is 0.261538

Chips 9 18 3, P is 0.205391, mal1 is 0.222222, mal2 is 0.216667

Chips 9 19 2, P is 0.147712, mal1 is 0.167401, mal2 is 0.160390

Chips 9 20 1, P is 0.079488, mal1 is 0.095694, mal2 is 0.089655

Chips 10 1 19, P is 0.069745, mal1 is 0.086758, mal2 is 0.079415

Chips 10 2 18, P is 0.129935, mal1 is 0.152542, mal2 is 0.142857

Chips 10 3 17, P is 0.181052, mal1 is 0.203187, mal2 is 0.193732

Chips 10 4 16, P is 0.223714, mal1 is 0.242424, mal2 is 0.234432

Chips 10 5 15, P is 0.258564, mal1 is 0.272727, mal2 is 0.266667

Chips 10 6 14, P is 0.286204, mal1 is 0.295775, mal2 is 0.291667

Chips 10 7 13, P is 0.307150, mal1 is 0.312715, mal2 is 0.310315

Chips 10 8 12, P is 0.321814, mal1 is 0.324324, mal2 is 0.323232

Chips 10 9 11, P is 0.330494, mal1 is 0.331104, mal2 is 0.330827

Chips 10 10 10, P is 0.333367, mal1 is 0.333333, mal2 is 0.333333

Chips 10 11 9, P is 0.330493, mal1 is 0.331104, mal2 is 0.330827

Chips 10 12 8, P is 0.321813, mal1 is 0.324324, mal2 is 0.323232

Chips 10 13 7, P is 0.307149, mal1 is 0.312715, mal2 is 0.310315

Chips 10 14 6, P is 0.286202, mal1 is 0.295775, mal2 is 0.291667

Chips 10 15 5, P is 0.258562, mal1 is 0.272727, mal2 is 0.266667

Chips 10 16 4, P is 0.223712, mal1 is 0.242424, mal2 is 0.234432

Chips 10 17 3, P is 0.181051, mal1 is 0.203187, mal2 is 0.193732

Chips 10 18 2, P is 0.129934, mal1 is 0.152542, mal2 is 0.142857

Chips 10 19 1, P is 0.069744, mal1 is 0.086758, mal2 is 0.079415

Chips 11 1 18, P is 0.061331, mal1 is 0.079295, mal2 is 0.070690

Chips 11 2 17, P is 0.114376, mal1 is 0.139918, mal2 is 0.127656

Chips 11 3 16, P is 0.159464, mal1 is 0.186770, mal2 is 0.173545

Chips 11 4 15, P is 0.197022, mal1 is 0.223048, mal2 is 0.210256

Chips 11 5 14, P is 0.227501, mal1 is 0.250896, mal2 is 0.239167

Chips 11 6 13, P is 0.251326, mal1 is 0.271777, mal2 is 0.261275

Chips 11 7 12, P is 0.268860, mal1 is 0.286689, mal2 is 0.277295

Chips 11 8 11, P is 0.280387, mal1 is 0.296296, mal2 is 0.287719

Chips 11 9 10, P is 0.286102, mal1 is 0.301003, mal2 is 0.292857

Chips 11 10 9, P is 0.286102, mal1 is 0.301003, mal2 is 0.292857

Chips 11 11 8, P is 0.280387, mal1 is 0.296296, mal2 is 0.287719

Chips 11 12 7, P is 0.268858, mal1 is 0.286689, mal2 is 0.277295

Chips 11 13 6, P is 0.251324, mal1 is 0.271777, mal2 is 0.261275

Chips 11 14 5, P is 0.227499, mal1 is 0.250896, mal2 is 0.239167

Chips 11 15 4, P is 0.197020, mal1 is 0.223048, mal2 is 0.210256

Chips 11 16 3, P is 0.159462, mal1 is 0.186770, mal2 is 0.173545

Chips 11 17 2, P is 0.114375, mal1 is 0.139918, mal2 is 0.127656

Chips 11 18 1, P is 0.061330, mal1 is 0.079295, mal2 is 0.070690

Chips 12 1 17, P is 0.053930, mal1 is 0.072961, mal2 is 0.063130

Chips 12 2 16, P is 0.100545, mal1 is 0.129032, mal2 is 0.114286

Chips 12 3 15, P is 0.140073, mal1 is 0.172414, mal2 is 0.155556

Chips 12 4 14, P is 0.172812, mal1 is 0.205882, mal2 is 0.188462

Chips 12 5 13, P is 0.199076, mal1 is 0.231317, mal2 is 0.214118

Chips 12 6 12, P is 0.219161, mal1 is 0.250000, mal2 is 0.233333

Chips 12 7 11, P is 0.233318, mal1 is 0.262799, mal2 is 0.246682

Chips 12 8 10, P is 0.241734, mal1 is 0.270270, mal2 is 0.254545

Chips 12 9 9, P is 0.244526, mal1 is 0.272727, mal2 is 0.257143

Chips 12 10 8, P is 0.241734, mal1 is 0.270270, mal2 is 0.254545

Chips 12 11 7, P is 0.233317, mal1 is 0.262799, mal2 is 0.246682

Chips 12 12 6, P is 0.219159, mal1 is 0.250000, mal2 is 0.233333

Chips 12 13 5, P is 0.199074, mal1 is 0.231317, mal2 is 0.214118

Chips 12 14 4, P is 0.172810, mal1 is 0.205882, mal2 is 0.188462

Chips 12 15 3, P is 0.140072, mal1 is 0.172414, mal2 is 0.155556

Chips 12 16 2, P is 0.100544, mal1 is 0.129032, mal2 is 0.114286

Chips 12 17 1, P is 0.053929, mal1 is 0.072961, mal2 is 0.063130

Chips 13 1 16, P is 0.047324, mal1 is 0.067511, mal2 is 0.056486

Chips 13 2 15, P is 0.088103, mal1 is 0.119522, mal2 is 0.102381

Chips 13 3 14, P is 0.122494, mal1 is 0.159696, mal2 is 0.139352

Chips 13 4 13, P is 0.150703, mal1 is 0.190476, mal2 is 0.168627

Chips 13 5 12, P is 0.172949, mal1 is 0.213523, mal2 is 0.191111

Chips 13 6 11, P is 0.189435, mal1 is 0.229965, mal2 is 0.207456

Chips 13 7 10, P is 0.200327, mal1 is 0.240550, mal2 is 0.218116

Chips 13 8 9, P is 0.205742, mal1 is 0.245734, mal2 is 0.223377

Chips 13 9 8, P is 0.205742, mal1 is 0.245734, mal2 is 0.223377

Chips 13 10 7, P is 0.200326, mal1 is 0.240550, mal2 is 0.218116

Chips 13 11 6, P is 0.189434, mal1 is 0.229965, mal2 is 0.207456

Chips 13 12 5, P is 0.172948, mal1 is 0.213523, mal2 is 0.191111

Chips 13 13 4, P is 0.150701, mal1 is 0.190476, mal2 is 0.168627

Chips 13 14 3, P is 0.122492, mal1 is 0.159696, mal2 is 0.139352

Chips 13 15 2, P is 0.088102, mal1 is 0.119522, mal2 is 0.102381

Chips 13 16 1, P is 0.047323, mal1 is 0.067511, mal2 is 0.056486

Chips 14 1 15, P is 0.041366, mal1 is 0.062762, mal2 is 0.050575

Chips 14 2 14, P is 0.076815, mal1 is 0.111111, mal2 is 0.091667

Chips 14 3 13, P is 0.106455, mal1 is 0.148289, mal2 is 0.124619

Chips 14 4 12, P is 0.130430, mal1 is 0.176471, mal2 is 0.150427

Chips 14 5 11, P is 0.148888, mal1 is 0.197133, mal2 is 0.169825

Chips 14 6 10, P is 0.161964, mal1 is 0.211268, mal2 is 0.183333

Chips 14 7 9, P is 0.169765, mal1 is 0.219512, mal2 is 0.191304

Chips 14 8 8, P is 0.172357, mal1 is 0.222222, mal2 is 0.193939

Chips 14 9 7, P is 0.169764, mal1 is 0.219512, mal2 is 0.191304

Chips 14 10 6, P is 0.161963, mal1 is 0.211268, mal2 is 0.183333

Chips 14 11 5, P is 0.148887, mal1 is 0.197133, mal2 is 0.169825

Chips 14 12 4, P is 0.130429, mal1 is 0.176471, mal2 is 0.150427

Chips 14 13 3, P is 0.106454, mal1 is 0.148289, mal2 is 0.124619

Chips 14 14 2, P is 0.076813, mal1 is 0.111111, mal2 is 0.091667

Chips 14 15 1, P is 0.041365, mal1 is 0.062762, mal2 is 0.050575

Chips 15 1 14, P is 0.035951, mal1 is 0.058577, mal2 is 0.045259

Chips 15 2 13, P is 0.066517, mal1 is 0.103586, mal2 is 0.081933

Chips 15 3 12, P is 0.091772, mal1 is 0.137931, mal2 is 0.111111

Chips 15 4 11, P is 0.111812, mal1 is 0.163569, mal2 is 0.133603

Chips 15 5 10, P is 0.126737, mal1 is 0.181818, mal2 is 0.150000

Chips 15 6 9, P is 0.136633, mal1 is 0.193548, mal2 is 0.160714

Chips 15 7 8, P is 0.141564, mal1 is 0.199288, mal2 is 0.166008

Chips 15 8 7, P is 0.141563, mal1 is 0.199288, mal2 is 0.166008

Chips 15 9 6, P is 0.136632, mal1 is 0.193548, mal2 is 0.160714

Chips 15 10 5, P is 0.126736, mal1 is 0.181818, mal2 is 0.150000

Chips 15 11 4, P is 0.111811, mal1 is 0.163569, mal2 is 0.133603

Chips 15 12 3, P is 0.091771, mal1 is 0.137931, mal2 is 0.111111

Chips 15 13 2, P is 0.066516, mal1 is 0.103586, mal2 is 0.081933

Chips 15 14 1, P is 0.035951, mal1 is 0.058577, mal2 is 0.045259

Chips 16 1 13, P is 0.031011, mal1 is 0.054852, mal2 is 0.040433

Chips 16 2 12, P is 0.057100, mal1 is 0.096774, mal2 is 0.073016

Chips 16 3 11, P is 0.078317, mal1 is 0.128405, mal2 is 0.098635

Chips 16 4 10, P is 0.094727, mal1 is 0.151515, mal2 is 0.117949

Chips 16 5 9, P is 0.106394, mal1 is 0.167286, mal2 is 0.131429

Chips 16 6 8, P is 0.113370, mal1 is 0.176471, mal2 is 0.139394

Chips 16 7 7, P is 0.115691, mal1 is 0.179487, mal2 is 0.142029

Chips 16 8 6, P is 0.113370, mal1 is 0.176471, mal2 is 0.139394

Chips 16 9 5, P is 0.106393, mal1 is 0.167286, mal2 is 0.131429

Chips 16 10 4, P is 0.094726, mal1 is 0.151515, mal2 is 0.117949

Chips 16 11 3, P is 0.078316, mal1 is 0.128405, mal2 is 0.098635

Chips 16 12 2, P is 0.057099, mal1 is 0.096774, mal2 is 0.073016

Chips 16 13 1, P is 0.031010, mal1 is 0.054852, mal2 is 0.040433

Chips 17 1 12, P is 0.026494, mal1 is 0.051502, mal2 is 0.036015

Chips 17 2 11, P is 0.048485, mal1 is 0.090535, mal2 is 0.064787

Chips 17 3 10, P is 0.066006, mal1 is 0.119522, mal2 is 0.087037

Chips 17 4 9, P is 0.079097, mal1 is 0.140078, mal2 is 0.103297

Chips 17 5 8, P is 0.087798, mal1 is 0.153257, mal2 is 0.113939

Chips 17 6 7, P is 0.092140, mal1 is 0.159696, mal2 is 0.119203

Chips 17 7 6, P is 0.092140, mal1 is 0.159696, mal2 is 0.119203

Chips 17 8 5, P is 0.087797, mal1 is 0.153257, mal2 is 0.113939

Chips 17 9 4, P is 0.079096, mal1 is 0.140078, mal2 is 0.103297

Chips 17 10 3, P is 0.066005, mal1 is 0.119522, mal2 is 0.087037

Chips 17 11 2, P is 0.048484, mal1 is 0.090535, mal2 is 0.064787

Chips 17 12 1, P is 0.026494, mal1 is 0.051502, mal2 is 0.036015

Chips 18 1 11, P is 0.022370, mal1 is 0.048458, mal2 is 0.031942

Chips 18 2 10, P is 0.040624, mal1 is 0.084746, mal2 is 0.057143

Chips 18 3 9, P is 0.054782, mal1 is 0.111111, mal2 is 0.076190

Chips 18 4 8, P is 0.064871, mal1 is 0.129032, mal2 is 0.089510

Chips 18 5 7, P is 0.070914, mal1 is 0.139442, mal2 is 0.097391

Chips 18 6 6, P is 0.072926, mal1 is 0.142857, mal2 is 0.100000

Chips 18 7 5, P is 0.070914, mal1 is 0.139442, mal2 is 0.097391

Chips 18 8 4, P is 0.064871, mal1 is 0.129032, mal2 is 0.089510

Chips 18 9 3, P is 0.054782, mal1 is 0.111111, mal2 is 0.076190

Chips 18 10 2, P is 0.040623, mal1 is 0.084746, mal2 is 0.057143

Chips 18 11 1, P is 0.022369, mal1 is 0.048458, mal2 is 0.031942

Chips 19 1 10, P is 0.018616, mal1 is 0.045662, mal2 is 0.028161

Chips 19 2 9, P is 0.033483, mal1 is 0.079295, mal2 is 0.050000

Chips 19 3 8, P is 0.044613, mal1 is 0.103004, mal2 is 0.065993

Chips 19 4 7, P is 0.052023, mal1 is 0.118143, mal2 is 0.076477

Chips 19 5 6, P is 0.055724, mal1 is 0.125523, mal2 is 0.081667

Chips 19 6 5, P is 0.055724, mal1 is 0.125523, mal2 is 0.081667

Chips 19 7 4, P is 0.052023, mal1 is 0.118143, mal2 is 0.076477

Chips 19 8 3, P is 0.044613, mal1 is 0.103004, mal2 is 0.065993

Chips 19 9 2, P is 0.033482, mal1 is 0.079295, mal2 is 0.050000

Chips 19 10 1, P is 0.018615, mal1 is 0.045662, mal2 is 0.028161

Chips 20 1 9, P is 0.015218, mal1 is 0.043062, mal2 is 0.024631

Chips 20 2 8, P is 0.027042, mal1 is 0.074074, mal2 is 0.043290

Chips 20 3 7, P is 0.035478, mal1 is 0.095023, mal2 is 0.056361

Chips 20 4 6, P is 0.040536, mal1 is 0.107143, mal2 is 0.064103

Chips 20 5 5, P is 0.042221, mal1 is 0.111111, mal2 is 0.066667

Chips 20 6 4, P is 0.040536, mal1 is 0.107143, mal2 is 0.064103

Chips 20 7 3, P is 0.035478, mal1 is 0.095023, mal2 is 0.056361

Chips 20 8 2, P is 0.027041, mal1 is 0.074074, mal2 is 0.043290

Chips 20 9 1, P is 0.015218, mal1 is 0.043062, mal2 is 0.024631

Chips 21 1 8, P is 0.012169, mal1 is 0.040609, mal2 is 0.021317

Chips 21 2 7, P is 0.021289, mal1 is 0.068966, mal2 is 0.036957

Chips 21 3 6, P is 0.027366, mal1 is 0.086957, mal2 is 0.047222

Chips 21 4 5, P is 0.030403, mal1 is 0.095694, mal2 is 0.052308

Chips 21 5 4, P is 0.030403, mal1 is 0.095694, mal2 is 0.052308

Chips 21 6 3, P is 0.027366, mal1 is 0.086957, mal2 is 0.047222

Chips 21 7 2, P is 0.021289, mal1 is 0.068966, mal2 is 0.036957

Chips 21 8 1, P is 0.012168, mal1 is 0.040609, mal2 is 0.021317

Chips 22 1 7, P is 0.009462, mal1 is 0.038251, mal2 is 0.018191

Chips 22 2 6, P is 0.016218, mal1 is 0.063830, mal2 is 0.030952

Chips 22 3 5, P is 0.020270, mal1 is 0.078534, mal2 is 0.038519

Chips 22 4 4, P is 0.021621, mal1 is 0.083333, mal2 is 0.041026

Chips 22 5 3, P is 0.020270, mal1 is 0.078534, mal2 is 0.038519

Chips 22 6 2, P is 0.016218, mal1 is 0.063830, mal2 is 0.030952

Chips 22 7 1, P is 0.009462, mal1 is 0.038251, mal2 is 0.018191

Chips 23 1 6, P is 0.007095, mal1 is 0.035928, mal2 is 0.015230

Chips 23 2 5, P is 0.011825, mal1 is 0.058480, mal2 is 0.025238

Chips 23 3 4, P is 0.014189, mal1 is 0.069364, mal2 is 0.030199

Chips 23 4 3, P is 0.014189, mal1 is 0.069364, mal2 is 0.030199

Chips 23 5 2, P is 0.011825, mal1 is 0.058480, mal2 is 0.025238

Chips 23 6 1, P is 0.007095, mal1 is 0.035928, mal2 is 0.015230

Chips 24 1 5, P is 0.005068, mal1 is 0.033557, mal2 is 0.012414

Chips 24 2 4, P is 0.008108, mal1 is 0.052632, mal2 is 0.019780

Chips 24 3 3, P is 0.009122, mal1 is 0.058824, mal2 is 0.022222

Chips 24 4 2, P is 0.008108, mal1 is 0.052632, mal2 is 0.019780

Chips 24 5 1, P is 0.005068, mal1 is 0.033557, mal2 is 0.012414

Chips 25 1 4, P is 0.003379, mal1 is 0.031008, mal2 is 0.009726

Chips 25 2 3, P is 0.005068, mal1 is 0.045802, mal2 is 0.014550

Chips 25 3 2, P is 0.005068, mal1 is 0.045802, mal2 is 0.014550

Chips 25 4 1, P is 0.003379, mal1 is 0.031008, mal2 is 0.009726

Chips 26 1 3, P is 0.002027, mal1 is 0.028037, mal2 is 0.007152

Chips 26 2 2, P is 0.002703, mal1 is 0.037037, mal2 is 0.009524

Chips 26 3 1, P is 0.002027, mal1 is 0.028037, mal2 is 0.007152

Chips 27 1 2, P is 0.001014, mal1 is 0.024096, mal2 is 0.004680

Chips 27 2 1, P is 0.001014, mal1 is 0.024096, mal2 is 0.004680

Chips 28 1 1, P is 0.000338, mal1 is 0.017544, mal2 is 0.002299

Below is a link to a post I made on rgp in 1997 describing this very process and solving it algebraically for the simplest case. I didn't take it much further, but it is interesting to note the importance of the granularity: It turns out it is not enough to just know the ratio of stack sizes (as I'm sure Bill's data demonstrates), but the actual number of bets in each stack is important.


Taking the granularity to be infinitesimal gives the differential equation which (along with the boundary conditions) provides the "small bet size approximation".


Tom Weideman