Pai Gow Tiles Statistics

I was bored this morning so I decided I wanted to crunch some numbers with regards to Pai Gow Tiles. I'm not a math guy -- this was something I did for fun, so it's entirely possible some of the numbers don't add up right and/or I just don't know what the heck i'm talking about. I based all of this off of information I got out of Musante's book on Pai Gow.

First of all, here are your chances of being able to make a specific total. I.e., if you are given two Pai Gow tiles at random, this is the percent chance of what the total will be:

Pair -- 7.24%
Wong -- 1.8%
Gong -- 2.71%
Nine -- 9.5%
Eight -- 9.95%
Seven -- 10.86%
Six -- 9.95%
Five -- 10.4%
Four -- 10.4%
Three -- 10.86%
Two -- 7.69%
One -- 5.43%
Zero -- 3.17%

Statistically, the two most likely totals are Three and Seven (by a hair), and the two least likely totals are Wong and Gong.

Next, here are your chances of winning with a specific total if you are holding the worst possible hand for that total. For example, if you are holding the worst possible pair, i.e., Chop Ng, that total beats 93.21% of all other totals. Another way to look at these numbers is to think, "Okay, these two tiles make a seven.... what are the absolute worst odds I have for that seven winning the hand?" If you have any given seven, that beats at least 58.37% of all other hands.

Pair -- 93.21%
Wong -- 91.4%
Gong -- 88.69%
Nine -- 79.19%
Eight -- 69.23%
Seven -- 58.37%
Six -- 48.42%
Five -- 38.01%
Four -- 27.6%
Three -- 16.74%
Two -- 9.05%
One -- 3.62%
Zero -- 0%

These numbers can be rounded off and boiled down into a handy little rule of thumb if you want to do some calculations on the fly at the gaming table: "Roughly speaking, what are the worst possible chances of this hand winning?"

Pair -- 95%
Wong & Gong -- 90%
Nine -- 80%
Eight -- 70%
Seven -- 60%
Six -- 50%
Five -- 40%
Four -- 30%
Three -- 15%
Two -- 10%
One -- 5%
Zero -- 0%

Okay, now tell me what I did wrong..... :D

I'm not sure about your figures since the dealer plays four tiles; also with just two tiles your chance of a pair are clearly 1 in 31 (you have a tile, is the second one its match or not).

These are the ballpark figures I get for the player playing an optimum strategy, remembering each combination creates two hands. However, as you can see from the wizard site, the numbers for the high and low hands are different and so are their chances of winning with a specific total.

Desc	Number	%
PAIR	6 328	8.799%
Wong	1 800	2.503%
Gong	3 480	4.839%
Nine	11 552	16.062%
Eight	9 000	12.514%
Seven	10 472	14.561%
Six	8 848	12.303%
Five	5 648	7.853%
Four	3 112	4.327%
Three	2 744	3.815%
Two	2 692	3.743%
One	3 752	5.217%
Zero	2 492	3.465%

Low	Number	%	High	Number	%
PAIR	120	0.334%	PAIR	6 208	17.264%
Wong	116	0.323%	Wong	1 684	4.683%
Gong	296	0.823%	Gong	3 184	8.854%
Nine	2 248	6.251%	Nine	9 304	25.873%
Eight	3 432	9.544%	Eight	5 568	15.484%
Seven	5 548	15.428%	Seven	4 924	13.693%
Six	6 288	17.486%	Six	2 560	7.119%
Five	4 688	13.037%	Five	960	2.670%
Four	2 408	6.696%	Four	704	1.958%
Three	2 088	5.806%	Three	656	1.824%
Two	2 484	6.908%	Two	208	0.578%
One	3 752	10.434%	One	0	0.000%
Zero	2 492	6.930%	Zero	0	0.000%

Quote: TigerWu

Okay, now tell me what I did wrong..... :D

Quick answer--- Dealer gets 4 tiles to make a hand, not 2.

Well, I'll tell you how I came up with my figures... In the Musante book he charts out all the possible tile combinations. I just counted them up and started crunching percentages. It's possible I just counted something wrong, or Musante didn't cover every possible angle, or, like I said, I just don't know what I'm doing mathematically. I made a few more charts of combinations just for fun, but obviously all this stuff has already been figured out, and probably much more accurately.

Thanks!

There are 31*31/2 combinations of two tiles i.e. 496. If you were to consider (say) High 8 (4-4) with a Low 8 (6-2 or 5-3) as a pair then you would get 36 combinations - and 36/496 = 7.26%, whereas normal Pai Gow would treat it as a six. Also, if using two tiles, a Wong should be half the chance of a Gong - since there are two 9-tiles and four 8-tiles to go with either a Teen or Day.
It's possible you added them up wrong - for instance there is only one way to get any specific pair whereas there are four ways to get a specific couple (e.g. Hi Wong:6-6a & 5-4, 6-6a & 6-3, 6-6b & 5-4, 6-6b & 5-4). So, in total, there are 8 ways to get a Wong 6-6 and 1-1 four ways each.

Quote: charliepatrick

It's possible you added them up wrong

Okay, yep, I know what I did wrong now. Thanks.

That's why I have an art degree and don't gamble professionally. :D