COSMICOOMPH
COSMICOOMPH
Joined: Feb 25, 2013
  • Threads: 1
  • 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
Paigowdan
Joined: Apr 28, 2010
  • Threads: 115
  • 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
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
February 25th, 2013 at 3:09:40 PM permalink
You mean played rounds

I think I read your question differently from Dan.
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%

added: other intervals
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)
added: complete data
Round # / Probability of it happening exactly on the Nth round / Cumulative (N or less)

1	0.017642699	0.017642699
2 0.017331434 0.034974134
3 0.017025661 0.051999795
4 0.016725282 0.068725077
5 0.016430203 0.085155281
6 0.01614033 0.101295611
7 0.015855571 0.117151182
8 0.015575836 0.132727018
9 0.015301036 0.148028055
10 0.015031085 0.163059139
11 0.014765896 0.177825035
12 0.014505386 0.192330421
13 0.014249471 0.206579892
14 0.013998072 0.220577965
15 0.013751109 0.234329073
16 0.013508502 0.247837575
17 0.013270175 0.261107751
18 0.013036054 0.274143804
19 0.012806063 0.286949867
20 0.012580129 0.299529996
21 0.012358182 0.311888178
22 0.01214015 0.324028327
23 0.011925965 0.335954292
24 0.011715559 0.347669851
25 0.011508865 0.359178716
26 0.011305817 0.370484533
27 0.011106352 0.381590885
28 0.010910406 0.392501291
29 0.010717917 0.403219208
30 0.010528824 0.413748032
31 0.010343067 0.424091099
32 0.010160588 0.434251687
33 0.009981327 0.444233014
34 0.00980523 0.454038244
35 0.009632239 0.463670483
36 0.0094623 0.473132783
37 0.00929536 0.482428143
38 0.009131365 0.491559508
39 0.008970263 0.50052977
40 0.008812003 0.509341773
41 0.008656536 0.517998309
42 0.008503811 0.52650212
43 0.008353781 0.5348559
44 0.008206397 0.543062298
45 0.008061614 0.551123912
46 0.007919386 0.559043298
47 0.007779666 0.566822965
48 0.007642412 0.574465377
49 0.007507579 0.581972956
50 0.007375125 0.589348082
51 0.007245008 0.59659309
52 0.007117187 0.603710277
53 0.00699162 0.610701897
54 0.006868269 0.617570166
55 0.006747095 0.624317261
56 0.006628058 0.630945318
57 0.006511121 0.637456439
58 0.006396247 0.643852686
59 0.0062834 0.650136086
60 0.006172544 0.65630863
61 0.006063643 0.662372273
62 0.005956664 0.668328938
63 0.005851573 0.674180511
64 0.005748335 0.679928846
65 0.005646919 0.685575765
66 0.005547292 0.691123057
67 0.005449423 0.69657248
68 0.00535328 0.701925761
69 0.005258834 0.707184595
70 0.005166054 0.712350649
71 0.005074911 0.71742556
72 0.004985376 0.722410936
73 0.00489742 0.727308356
74 0.004811017 0.732119373
75 0.004726137 0.73684551
76 0.004642756 0.741488266
77 0.004560845 0.746049111
78 0.004480379 0.75052949
79 0.004401333 0.754930823
80 0.004323682 0.759254505
81 0.0042474 0.763501905
82 0.004172465 0.76767437
83 0.004098851 0.771773221
84 0.004026536 0.775799757
85 0.003955497 0.779755255
86 0.003885712 0.783640967
87 0.003817157 0.787458124
88 0.003749812 0.791207936
89 0.003683656 0.794891592
90 0.003618666 0.798510258
91 0.003554823 0.802065081
92 0.003492106 0.805557187
93 0.003430496 0.808987683
94 0.003369973 0.812357656
95 0.003310517 0.815668173
96 0.003252111 0.818920284
97 0.003194735 0.822115019
98 0.003138371 0.825253391
99 0.003083002 0.828336392
100 0.003028609 0.831365002

Here 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 0.17166361
100 0.00302861 0.82833639 0.17166361 0.83136500 0.16863500


Hope this can help someone
winsome johnny (not Win some johnny)
COSMICOOMPH
COSMICOOMPH
Joined: Feb 25, 2013
  • Threads: 1
  • 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.

Thank you for your input.
COSMICOOMPH
COSMICOOMPH
Joined: Feb 25, 2013
  • Threads: 1
  • 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
rdw4potus
Joined: Mar 11, 2010
  • Threads: 80
  • 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
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • 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.

Thank you for your input.

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.
(I answered your SD question in your blog)

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
COSMICOOMPH
Joined: Feb 25, 2013
  • Threads: 1
  • 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
rdw4potus
Joined: Mar 11, 2010
  • Threads: 80
  • 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
Gabes22
Joined: Jul 19, 2011
  • Threads: 15
  • Posts: 1427
February 26th, 2013 at 6:12:56 AM permalink
The odds of it happening between hands 52 and 62 are the same as it happening from hands 1-11
The odds of it happening from hands 47 to 67 are the same as it happening from hands 1-21
The odds of it happening from hands 42 to 72 are the same as it happening from hands 1-31.

The only way to get an edge on a 50:1 payout which has a 57:1 chance is all about fortuitous timing, i.e. dumb luck
A flute with no holes is not a flute, a donut with no holes is a danish

  • Jump to: