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

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.

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

Thank you for your input.

Thanks.

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...

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

(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

This is also known as the wait time distribution

Cumulative, for X or less plays

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

You mean average.Quote:COSMICOOMPHJust by deduction, that tends me to think that cannot be correct and still have a median of 57 hands.

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

Yep. You are mixing up "averages" and "probabilities"Quote:COSMICOOMPHAm I missing something?

57 will not be the apex of the bell curve.Quote:COSMICOOMPHI 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.

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

Quote:COSMICOOMPHI 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.

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