COSMICOOMPH
Joined: Feb 25, 2013
• Posts: 4
February 25th, 2013 at 2:18:40 PM permalink
A queen high pai gow occurs about every 57 hands. My question is this: What is the percentage of times it will occur between the following set of played hands:

52 to 62 hands: ____%

47 to 67 hands: ____%

42 to 72 hands ____%

The reason I ask is if the house pays 50 to 1 odds on a queen high, I would like to play the statistics in the safest range.

Thanks.

cosmicoomph
Paigowdan
Joined: Apr 28, 2010
• Posts: 5692
February 25th, 2013 at 2:40:41 PM permalink
Yes, a dealer's queen high hand occurs about once in every 57 rounds or so.

Since the cards are reshuffled and re-dealt for each round of play, any round of play has exactly the same odds of the queen's dragon bet hitting, which is 1.7643%.

In 11 rounds of play, you will see - on average - a dealer's queen's dragon hand occuring once in every 5.153 groups of 11 rounds.

in 21 rounds of play, you see it once in every 2.69 groups of 21 rounds, on average.

In 31 rounds of play, you'll see it once in every 1.828 groups of 31 rounds, on average.

In 57 rounds of play, you'll see it, on average, once in a group of 57 hands.
Beware of all enterprises that require new clothes - Henry David Thoreau. Like Dealers' uniforms - Dan.
7craps
Joined: Jan 23, 2010
• Posts: 1977
February 25th, 2013 at 3:09:40 PM permalink
You mean played rounds

You want the probabilities and not the averages.
Well, you got both.

I used 19425/1101022
for the probability from the Wizard's site

Including the rounds a to b (example 52 to 62)

52-62: 7.1736%
47-67: 13.7529%
42-72: 20.4413%

1-39: 50.052977% (this is the median. half by the 39th round, the other half after)
1-50: 58.9348082%

You do not want to know how to do this yourself??
It is very simple in a spreadsheet, more work with just a pencil and paper and calculator,
but making a table of values and adding them up is a simple task.

P= 19425/1101022 = probability of success
Q= 1-P
N = the round number

For it to happen on the first round (hand)
P

2nd round
Q*P (It did not happen on the first round but did happen on the 2nd round)

3rd round
Q*Q*P (Q^2*P)
4th round
Q*Q*Q*P (Q^3*P)
and so on

a pattern develops
Q^(N-1)*P

We can easily fill in our table
Then just add up the intervals (say 52 to 62) you desire to include.

There are a few other ways to do this, but I find this easy in a spreadsheet
here is a table for the first 100 rounds (dealer hands)
Round # / Probability of it happening exactly on the Nth round / Cumulative (N or less)

`1	0.017642699	0.0176426992	0.017331434	0.0349741343	0.017025661	0.0519997954	0.016725282	0.0687250775	0.016430203	0.0851552816	0.01614033	0.1012956117	0.015855571	0.1171511828	0.015575836	0.1327270189	0.015301036	0.14802805510	0.015031085	0.16305913911	0.014765896	0.17782503512	0.014505386	0.19233042113	0.014249471	0.20657989214	0.013998072	0.22057796515	0.013751109	0.23432907316	0.013508502	0.24783757517	0.013270175	0.26110775118	0.013036054	0.27414380419	0.012806063	0.28694986720	0.012580129	0.29952999621	0.012358182	0.31188817822	0.01214015	0.32402832723	0.011925965	0.33595429224	0.011715559	0.34766985125	0.011508865	0.35917871626	0.011305817	0.37048453327	0.011106352	0.38159088528	0.010910406	0.39250129129	0.010717917	0.40321920830	0.010528824	0.41374803231	0.010343067	0.42409109932	0.010160588	0.43425168733	0.009981327	0.44423301434	0.00980523	0.45403824435	0.009632239	0.46367048336	0.0094623	0.47313278337	0.00929536	0.48242814338	0.009131365	0.49155950839	0.008970263	0.5005297740	0.008812003	0.50934177341	0.008656536	0.51799830942	0.008503811	0.5265021243	0.008353781	0.534855944	0.008206397	0.54306229845	0.008061614	0.55112391246	0.007919386	0.55904329847	0.007779666	0.56682296548	0.007642412	0.57446537749	0.007507579	0.58197295650	0.007375125	0.58934808251	0.007245008	0.5965930952	0.007117187	0.60371027753	0.00699162	0.61070189754	0.006868269	0.61757016655	0.006747095	0.62431726156	0.006628058	0.63094531857	0.006511121	0.63745643958	0.006396247	0.64385268659	0.0062834	0.65013608660	0.006172544	0.6563086361	0.006063643	0.66237227362	0.005956664	0.66832893863	0.005851573	0.67418051164	0.005748335	0.67992884665	0.005646919	0.68557576566	0.005547292	0.69112305767	0.005449423	0.6965724868	0.00535328	0.70192576169	0.005258834	0.70718459570	0.005166054	0.71235064971	0.005074911	0.7174255672	0.004985376	0.72241093673	0.00489742	0.72730835674	0.004811017	0.73211937375	0.004726137	0.7368455176	0.004642756	0.74148826677	0.004560845	0.74604911178	0.004480379	0.7505294979	0.004401333	0.75493082380	0.004323682	0.75925450581	0.0042474	0.76350190582	0.004172465	0.7676743783	0.004098851	0.77177322184	0.004026536	0.77579975785	0.003955497	0.77975525586	0.003885712	0.78364096787	0.003817157	0.78745812488	0.003749812	0.79120793689	0.003683656	0.79489159290	0.003618666	0.79851025891	0.003554823	0.80206508192	0.003492106	0.80555718793	0.003430496	0.80898768394	0.003369973	0.81235765695	0.003310517	0.81566817396	0.003252111	0.81892028497	0.003194735	0.82211501998	0.003138371	0.82525339199	0.003083002	0.828336392100	0.003028609	0.831365002Here is a complete table of probabilities  up to 100 rounds  x    prob[X=x]   prob[X<x]  prob[X>=x]  prob[X<=x]   prob[X>x]  1   0.01764270  0.00000000  1.00000000  0.01764270  0.98235730  2   0.01733143  0.01764270  0.98235730  0.03497413  0.96502587  3   0.01702566  0.03497413  0.96502587  0.05199979  0.94800021  4   0.01672528  0.05199979  0.94800021  0.06872508  0.93127492  5   0.01643020  0.06872508  0.93127492  0.08515528  0.91484472  6   0.01614033  0.08515528  0.91484472  0.10129561  0.89870439  7   0.01585557  0.10129561  0.89870439  0.11715118  0.88284882  8   0.01557584  0.11715118  0.88284882  0.13272702  0.86727298  9   0.01530104  0.13272702  0.86727298  0.14802805  0.85197195 10   0.01503108  0.14802805  0.85197195  0.16305914  0.83694086 11   0.01476590  0.16305914  0.83694086  0.17782503  0.82217497 12   0.01450539  0.17782503  0.82217497  0.19233042  0.80766958 13   0.01424947  0.19233042  0.80766958  0.20657989  0.79342011 14   0.01399807  0.20657989  0.79342011  0.22057796  0.77942204 15   0.01375111  0.22057796  0.77942204  0.23432907  0.76567093 16   0.01350850  0.23432907  0.76567093  0.24783757  0.75216243 17   0.01327018  0.24783757  0.75216243  0.26110775  0.73889225 18   0.01303605  0.26110775  0.73889225  0.27414380  0.72585620 19   0.01280606  0.27414380  0.72585620  0.28694986  0.71305014 20   0.01258013  0.28694986  0.71305014  0.29952999  0.70047001 21   0.01235818  0.29952999  0.70047001  0.31188817  0.68811183 22   0.01214015  0.31188817  0.68811183  0.32402832  0.67597168 23   0.01192596  0.32402832  0.67597168  0.33595429  0.66404571 24   0.01171556  0.33595429  0.66404571  0.34766985  0.65233015 25   0.01150886  0.34766985  0.65233015  0.35917871  0.64082129 26   0.01130582  0.35917871  0.64082129  0.37048453  0.62951547 27   0.01110635  0.37048453  0.62951547  0.38159088  0.61840912 28   0.01091041  0.38159088  0.61840912  0.39250129  0.60749871 29   0.01071792  0.39250129  0.60749871  0.40321920  0.59678080 30   0.01052882  0.40321920  0.59678080  0.41374803  0.58625197 31   0.01034307  0.41374803  0.58625197  0.42409109  0.57590891 32   0.01016059  0.42409109  0.57590891  0.43425168  0.56574832 33   0.00998133  0.43425168  0.56574832  0.44423301  0.55576699 34   0.00980523  0.44423301  0.55576699  0.45403824  0.54596176 35   0.00963224  0.45403824  0.54596176  0.46367048  0.53632952 36   0.00946230  0.46367048  0.53632952  0.47313278  0.52686722 37   0.00929536  0.47313278  0.52686722  0.48242814  0.51757186 38   0.00913136  0.48242814  0.51757186  0.49155950  0.50844050 39   0.00897026  0.49155950  0.50844050  0.50052977  0.49947023 40   0.00881200  0.50052977  0.49947023  0.50934177  0.49065823 41   0.00865654  0.50934177  0.49065823  0.51799830  0.48200170 42   0.00850381  0.51799830  0.48200170  0.52650211  0.47349789 43   0.00835378  0.52650211  0.47349789  0.53485590  0.46514410 44   0.00820640  0.53485590  0.46514410  0.54306229  0.45693771 45   0.00806161  0.54306229  0.45693771  0.55112391  0.44887609 46   0.00791939  0.55112391  0.44887609  0.55904329  0.44095671 47   0.00777967  0.55904329  0.44095671  0.56682296  0.43317704 48   0.00764241  0.56682296  0.43317704  0.57446537  0.42553463 49   0.00750758  0.57446537  0.42553463  0.58197295  0.41802705 50   0.00737513  0.58197295  0.41802705  0.58934808  0.41065192 51   0.00724501  0.58934808  0.41065192  0.59659308  0.40340692 52   0.00711719  0.59659308  0.40340692  0.60371027  0.39628973 53   0.00699162  0.60371027  0.39628973  0.61070189  0.38929811 54   0.00686827  0.61070189  0.38929811  0.61757016  0.38242984 55   0.00674709  0.61757016  0.38242984  0.62431726  0.37568274 56   0.00662806  0.62431726  0.37568274  0.63094531  0.36905469 57   0.00651112  0.63094531  0.36905469  0.63745643  0.36254357 58   0.00639625  0.63745643  0.36254357  0.64385268  0.35614732 59   0.00628340  0.64385268  0.35614732  0.65013608  0.34986392 60   0.00617254  0.65013608  0.34986392  0.65630862  0.34369138 61   0.00606364  0.65630862  0.34369138  0.66237227  0.33762773 62   0.00595666  0.66237227  0.33762773  0.66832893  0.33167107 63   0.00585157  0.66832893  0.33167107  0.67418051  0.32581949 64   0.00574834  0.67418051  0.32581949  0.67992884  0.32007116 65   0.00564692  0.67992884  0.32007116  0.68557576  0.31442424 66   0.00554729  0.68557576  0.31442424  0.69112305  0.30887695 67   0.00544942  0.69112305  0.30887695  0.69657248  0.30342752 68   0.00535328  0.69657248  0.30342752  0.70192576  0.29807424 69   0.00525883  0.70192576  0.29807424  0.70718459  0.29281541 70   0.00516605  0.70718459  0.29281541  0.71235064  0.28764936 71   0.00507491  0.71235064  0.28764936  0.71742556  0.28257444 72   0.00498538  0.71742556  0.28257444  0.72241093  0.27758907 73   0.00489742  0.72241093  0.27758907  0.72730835  0.27269165 74   0.00481102  0.72730835  0.27269165  0.73211937  0.26788063 75   0.00472614  0.73211937  0.26788063  0.73684551  0.26315449 76   0.00464276  0.73684551  0.26315449  0.74148826  0.25851174 77   0.00456084  0.74148826  0.25851174  0.74604911  0.25395089 78   0.00448038  0.74604911  0.25395089  0.75052948  0.24947052 79   0.00440133  0.75052948  0.24947052  0.75493082  0.24506918 80   0.00432368  0.75493082  0.24506918  0.75925450  0.24074550 81   0.00424740  0.75925450  0.24074550  0.76350190  0.23649810 82   0.00417246  0.76350190  0.23649810  0.76767437  0.23232563 83   0.00409885  0.76767437  0.23232563  0.77177322  0.22822678 84   0.00402654  0.77177322  0.22822678  0.77579975  0.22420025 85   0.00395550  0.77579975  0.22420025  0.77975525  0.22024475 86   0.00388571  0.77975525  0.22024475  0.78364096  0.21635904 87   0.00381716  0.78364096  0.21635904  0.78745812  0.21254188 88   0.00374981  0.78745812  0.21254188  0.79120793  0.20879207 89   0.00368366  0.79120793  0.20879207  0.79489159  0.20510841 90   0.00361867  0.79489159  0.20510841  0.79851025  0.20148975 91   0.00355482  0.79851025  0.20148975  0.80206508  0.19793492 92   0.00349211  0.80206508  0.19793492  0.80555718  0.19444282 93   0.00343050  0.80555718  0.19444282  0.80898768  0.19101232 94   0.00336997  0.80898768  0.19101232  0.81235765  0.18764235 95   0.00331052  0.81235765  0.18764235  0.81566817  0.18433183 96   0.00325211  0.81566817  0.18433183  0.81892028  0.18107972 97   0.00319473  0.81892028  0.18107972  0.82211502  0.17788498 98   0.00313837  0.82211502  0.17788498  0.82525339  0.17474661 99   0.00308300  0.82525339  0.17474661  0.82833639  0.17166361100   0.00302861  0.82833639  0.17166361  0.83136500  0.16863500`

Hope this can help someone
winsome johnny (not Win some johnny)
COSMICOOMPH
Joined: Feb 25, 2013
• Posts: 4
February 25th, 2013 at 5:21:44 PM permalink
Thank you for your response and your work but I am not sure I am following. A queen high on an average happens every 57 or so rounds but yet your numbers show the probability of that happening before 57 plays is greater than after 57 plays and only about 20% between 42 and 72 plays. Just by deduction, that tends me to think that cannot be correct and still have a median of 57 hands. Am I missing something? I am thinking the percentage of a queen high between 42 and 72 hands should be nearer the 50% range if not higher and less before and after incrementally decreasing the further you are away from the median, unless of course it has not be hit by 57 hands then it would be increasing. But, we are looking for the bell curve of probability here with 57 being the apex of the curve and starting at 0 hands played.

COSMICOOMPH
Joined: Feb 25, 2013
• Posts: 4
February 25th, 2013 at 5:29:12 PM permalink
I do not think your answer is accurate. A queen high pai gow happens on average about every 57 hands played I believe per player (seems pretty accurate from my experience), not per a group of players, because if 7 were playing, a queen high would happen only every 399 individual hands played per your logic. Am I missing something here? Please regard my answer to the other individual that responded to get a better idea of what I am looking for. I am looking for the bell curve with 57 being the apex of the curve and wanting to know the probability in percentage of hitting the queen high between 52 and 62 hands, etc.. I am thinking that in the range of 42 and 72 hands it should be somewhere in the neighborhood of about 50% or so but do not know the math to prove it.

Thanks.
rdw4potus
Joined: Mar 11, 2010
• Posts: 7237
February 25th, 2013 at 6:35:50 PM permalink
Quote: COSMICOOMPH

The reason I ask is if the house pays 50 to 1 odds on a queen high, I would like to play the statistics in the safest range.

Thanks.

cosmicoomph

But this math doesn't affect the house edge in any way...
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
7craps
Joined: Jan 23, 2010
• Posts: 1977
February 25th, 2013 at 8:14:16 PM permalink
Quote: COSMICOOMPH

Thank you for your response and your work but I am not sure I am following.
A queen high on an average happens every 57 or so rounds but yet your numbers show the probability of that happening before 57 plays is greater than after 57 plays

Exactly and that is correct. The chance that it happens at least 1 time in 57 plays is ONLY 63.746%

(see my updated data table for the probabilities)
Here are the pics of the probability curves, Relative (for each Play) and Cumulative, for X or less plays
No peak around 57 plays.
This is also known as the wait time distribution

Cumulative, for X or less plays

Quote: COSMICOOMPH

and only about 20% between 42 and 72 plays.

Correct. A 20% probability of it happening between 42 and 72 plays inclusive.
Quote: COSMICOOMPH

Just by deduction, that tends me to think that cannot be correct and still have a median of 57 hands.

You mean average.
The median is the 50/50 or over/under value for all the possible outcomes in a finite number of trials.
The probabilities in the table are correct. They are probabilities and not averages.
You are confusing an average with a probability
Quote: COSMICOOMPH

Am I missing something?

Yep. You are mixing up "averages" and "probabilities"
Quote: COSMICOOMPH

I am thinking the percentage of a queen high between 42 and 72 hands should be nearer the 50% range if not higher and less before and after incrementally decreasing the further you are away from the median, unless of course it has not be hit by 57 hands then it would be increasing.
But, we are looking for the bell curve of probability here with 57 being the apex of the curve and starting at 0 hands played.

57 will not be the apex of the bell curve.
That would only happen at about 3231 plays
Again, you are mixing up and "average" and a "probability".
You need to review prob/stats 101. You got some ideas mixed up.

The "mean" or average can be in the center of the curve (also the mode and median)
IF and ONLY IF it is a normal distribution.
At 100 plays, the curve you want is NOT normal. Not even close.

The average comes from ALL the possible outcomes divided by the number of successes in N trials.
N/n
N = all possible outcomes (the sample space)
n= number of successes
A Probability has a value between 0 and 1 and is in the form of n/N

Histograms of 100 and 3231 plays
One looks normal and we can use the mean and standard deviation to find the probability of any interval
One is not and if we use the mean and sd we will not be happy with the results as explained in your blog

Basically, the "probability" of your event hitting on the (say)
32nd hand to the 56th hand is the total of all the individual probabilities of hitting on 32,33,...,55,56
forgot to add: when you have your probabilities (between 0 and 1) just multiply that value by 100 to get the % value

Sometimes these concepts of averages and probabilities can get confusing. Just remember which one you are dealing with
Good Luck
winsome johnny (not Win some johnny)
COSMICOOMPH
Joined: Feb 25, 2013
• Posts: 4
February 25th, 2013 at 8:52:00 PM permalink
I appreciate your response. I will add up some of your numbers and see if a bet 15 or 20 consecutively plays for a queen high on a 50 to 1 payout can pay for itself over time in a certain spread. If you have an answer to this, I would love to hear. The house always has numbers to its favor so it would be pleasant to hedge them whenever possible.
rdw4potus
Joined: Mar 11, 2010
• Posts: 7237
February 26th, 2013 at 6:04:37 AM permalink
Quote: COSMICOOMPH

I appreciate your response. I will add up some of your numbers and see if a bet 15 or 20 consecutively plays for a queen high on a 50 to 1 payout can pay for itself over time in a certain spread.

Of course it can't.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
Gabes22
Joined: Jul 19, 2011