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Ask the Slot Expert: Six royals in a cycle

16 March 2022

Question: I just finished what I consider to be a rather strange sequence of Royal Flushes. In my 35 years of playing Video Poker I have not come close to such a sequence.

I play two sessions a week at the Indian casino here in AZ. One is about 5 hours long and the other one 7 hours. I only play 8-6 Triple Double Bonus.

Over a very recent period of 12 sessions that, according to my records, covered approximately 57,000 hands played, I have had six Royals. The odds of that would seem to this Engineer to be very high. What do you think?

P.S. The casino here still has their mask policy and no smoking. Also, no eating at the machines.

Answer: What do I think? I think I need you to sit next to me the next time I play to see if some of your good luck will rub off onto me.

Twelve sessions at two session per week means you averaged a royal a week. At 57,000 hands played, let's round up and say you averaged 10,000 hands between royals.

I've had some royals hit pretty close together, but never a string of six hitting pretty much back-to-back-to-back-to-back-to-back-to-back. I did have 10 royals one day, but that was when I was dealt a royal on a 10-play machine. The time I hit two royals in one day was also done on a multi-hand machine, but on separate plays.

Baron Siméon Denis Poisson (1781-1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. (See how much you can accomplish when you don't have Facebook and Netflix.) We're interested in the work he did on random, independent events occurring in a fixed period of time.

Let's make a simplifying assumption so we can cheat off of the work others have already done. Let's assume that the number of hands you played is the royal flush cycle, the mean waiting time between royals. We can then look at the Wikipedia page for the Poisson Distribution to see the probability that an event will occur a certain number of times in one cycle.

The following table gives the probability that you will hit 0, 1, 2, etc. royals in a cycle.

0 - 0.368
1 - 0.368
2 - 0.184
3 - 0.061
4 - 0.015
5 - 0.003
6 - 0.0005

The first thing I noticed in the probabilities is that you're just as likely to get no royals in a cycle as one. I would have thought that hitting zero royals would have been less likely than hitting one, though the number of times I've played a cycle and hit no royals does argue that a zero-royal cycle isn't an uncommon occurrence.

Let's go down to the last line. The probability of hitting 6 royals in a cycle is 0.0005. Put another way, 1 cycle out of 2000 will contain 6 royals. If we use 40,000 hands per cycle, 2000 cycles is 80,000,000 hands. I don't think many players ever play that many hands in their careers.

Bask in the glory of your good fortune. You're part of an exclusive club.


It's interesting that your casino still requires masks. Although casinos (and any other businesses, for that matter) in Nevada are free to require them, none do. A few days ago, our positivity rate once again fell to about 5%, the level at which the pandemic is considered manageable. Our governor was a bit premature in lifting the mask mandate a month ago, but I guess we would have looked rather foolish with tens of thousands of mask-less visitors in Las Vegas for the Super Bowl ignoring the mandate.

It always seemed counterproductive to me that you were supposed to wear a mask on the gaming floor, but put an ashtray or drink in front of you and you were excused. Last year, I had to take a phone call when I was at a video poker machine. The person on the other end had difficulty understanding me, so I pulled down my mask. Twenty seconds later a guard walked by and motioned for me to put my mask on again. Never mind the people I could see smoking and drinking who hadn't had their masks on for 10-15 minutes.

It's been years since I've eaten at a machine. When I was a Las Vegas tourist, I would frequently get a bagel with cream cheese and an iced coffee for breakfast and enjoy them while playing a slot or video poker machine.

I haven't eaten at a machine since moving to Las Vegas. The closest thing would be picking up a breakfast sandwich and cold brew from the Dunkin Donuts near Red Rock Casino and then eating at the Onyx bar. I never left a mess behind, but I guess enough people did that they put up stanchions to keep people out when the bar was closed.

I don't mind wearing a mask. When I first moved to Las Vegas, I had a cough or cold practically every other month. I haven't had anything for a couple of years now. Even my cousin said that I used to get something a few times a year and it's been a long time since I've had a cold.

I still wear a mask in public places. I'm surprised at the number of people who have visible conditions that indicate they are more likely for bad Covid outcomes who don't wear one.

Here are the latest Covid data. There is a difference in the hospitalization data for US versus NV. The CDC reports total number of Covid hospital admissions. Nevada reports current hospitalizations, not admissions.

There must be something wrong with the latest case count from Nevada. Our test positivity rate has been steadily falling the past few weeks, but the number of cases reported this week is huge increase over the number of cases reported last week. The Nevada site has only 688,054 total cases as of 3/10, an increase that is more in line with the prior weeks' increases.

All data comes from the CDC (https://www.cdc.gov/covid-data-tracker/#cases_totalcases, https://covid.cdc.gov/covid-data-tracker/#new-hospital-admissions), except for Nevada current hospitalizations (https://nvhealthresponse.nv.gov/).


Totals Weekly Changes
US NV US NV
Date Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths  Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths 
 03/15/22   79,418,377   4,571,017   963,244   710,709   258   9,933   253,069   17,393   5,492   24,844   (20)   113 
 03/08/22   79,165,308   4,553,624   957,752   685,865   278   9,820   310,308   24,471   9,870   2,099   (101)   116 
 03/01/22   78,855,000   4,529,153   947,882   683,766   379   9,704   465,845   33,261   14,988   6,421   (189)   196 
 02/22/22   78,389,155   4,495,892   932,894   677,345   568   9,508   551,462   46,726   12,797   4,023   (297)   173 
 02/15/22   77,837,693   4,449,166   920,097   673,322   865   9,335   1,005,691   67,963   17,059   5,280   (375)   149 
 02/08/22   76,782,002   4,381,203   903,038   668,042   1,240   9,186   1,769,556   92,750   18,185   11,144   (387)   235 
 02/01/22   75,012,446   4,288,453   884,853   656,898   1,627   8,951   3,193,570   121,420   17,885   26,555   (344)   191 
 01/25/22   71,818,876   4,167,033   866,968   630,343   1,971   8,760   5,102,999   143,329   16,393   57,574   99   133 
 01/18/22   66,715,937   4,023,704   850,575   572,769   1,872   8,627   4,983,654   147,105   13,301   25,927   (1,787)   99 
 01/11/22   61,732,283   3,876,599   837,274   546,842   3,659   8,528   5,421,565   139,572   12,168   32,486   2,594   100 
 01/05/22   56,310,718   3,737,027   825,106   514,344   1,065   8,428   3,501,427   104,272   8,867   20,966   276   61 
 12/28/21   52,809,291   3,362,755   816,239   493,378   789   8,367   1,693,987   63,185   11,127   7,977   89   56 
 12/21   51,115,304   3,569,570   805,112   485,401   700   8,311   1,063,296   55,734   9,102   5,639   9   110 
 12/14   50,052,008   3,513,836   796,010   479,762   691   8,201   853,262   57,885   8,946   5,008   17   91 
 12/07   49,198,746   3,455,951   787,064   474,754   674   8,110   821,215   66,191   8,575   5,011   23   125 
 11/30   48,377,531   3,389,760   778,489   469,743   651   7,985   563,746   28,426   6,309   3,343   (50)   55 
 11/23   47,813,785   3,361,334   772,180   466,400   701   7,930   667,924   41,145   10,754   4,784   (24)   86 
 11/16   47,145,861   3,320,189   761,426   461,616   725   7,844   604,748   63,164   7,862   4,577   22   85 
 11/09   46,541,113   3,257,025   753,564   457,039   703   7,759   516,764   33,219   8,290   5,539   34   103 
 11/02   46,024,349   3,223,806   745,274   451,500   669   7,656   555,915   38,028   9,226   4,327   45   109 
 10/26   45,468,434   3,185,778   736,048   447,173   624   7,547   488,829   42,265   9,842   4,753   (87)   96 
 10/19   44,979,605   3,143,513   726,206   442,420   711   7,451   578,396   11,963   4,381   136 
 10/12   44,401,209   714,243   438,039   7,315   795,586   14,067   15,710   149 
 10/05   43,605,623   700,176   422,329   7,166   554,194   10,642   3,852   121 
 09/28   43,051,429   689,534   418,477   7,045   817,218   14,463   6,160   165 
 09/21   42,234,211   675,071   412,317   6,880   971,637   14,691   7,466   152 
 09/14   41,262,574   660,380   404,851   6,728   1,176,763   12,919   10,256   145 
 09/07   40,085,811   647,461   394,595   6,583   975,725   10,076   5,237   104 
 08/31   39,110,086   637,385   389,358   6,479   1,113,414   9,385   7,592   173 
 08/24   37,996,672   628,000   381,649   6,306   1,045,491   7,507   8,117   116 
 08/17   36,951,181   620,493   373,649   6,190   959,978   4,715   7,065   150 
 08/10   35,991,203   615,778   366,584   6,040   819,524   3,987   7,652   122 
 08/03   35,171,679   611,791   358,932   5,918   622,832   2,279   7,489   81 
 07/27   34,548,847   609,012   351,443   5,837   652,251   2,394   8,347   79 
 07/20   33,896,296   606,618   341,096   5,758   169,933   1,478   3,351   28 
 07/13   33,726,363   605,140   339,745   5,730   181,047   1,959   4,982   33 
 07/06   33,545,316   603,181   334,763   5,697   75,104   1,373   2,234   27 
 06/29   33,470,212   601,808   332,529   5,670   87,507   2,057   3,020   24 
 06/22   33,382,705   599,751   329,509   5,646   75,420   2,157   1,930   22 
 06/15   33,303,285   597,594   327,579   5,624   95,797   2,293   1,560   17 
 06/08   33,207,488   595,301   326,019   5,607   114,250   3,762   2,271   21 
 06/01   33,039,238   591,539   323,748   5,586   123,333   3,709   991   27 
 05/25   32,969,905   587,830   322,757   5,559   174,125   4,234   1,676   26 
 05/18   32,795,780   583,596   321,081   5,533   223,966   4,230   2,301   27 
 05/11   32,571,814   579,366   318,780   5,506   303,856   4,687   2,541   33 
 05/04   32,267,958   574,679   316,239   5,473   343,348   4,908   2,559   40 
 04/27   31,924,610   569,771   313,680   5,433   383,163   4,958   2,747   65 
 04/20   31,541,447   564,813   310,933   5,368   464,556   5,072   2,590   36 
 04/13   31,076,891   559,741   308,343   5,332   480,061   5,321   2,986   57 
 04/06   30,596,830   554,420   305,357   5,275   448,935   7,124   3,084   38 
 03/30   30,147,895   547,296   302,273   5,237   439,510   6,793   939   63 
 03/23   29,708,385   540,503   301,334   5,174   388,928   7,446   1,863   53 
 03/16   29,319,457   533,057   299,471   5,121   381,695   8,362   3,078   81 
 03/09   28,937,762   524,695   296,393   5,040   480,902   11,573   2,413   83 
 03/02   28,456,860   513,122   293,980   4,957   463,356   14,129   2,835   75 
 02/23   27,993,504   498,993   291,145   4,882   451,083   13,923   2,406   162 
 02/16   27,542,421   485,070   288,739   4,720   602,906   21,411   4,149   198 
 02/09   26,939,515   463,659   284,590   4,522   779,305   21,828   5,444   244 
 02/02   26,160,210   441,831   279,146   4,278   1,007,777   22,004   7,249   249 
 01/26   25,152,433   419,827   271,897   4,029   1,312,565   23,385   10,324   250 
 01/19   23,839,868   396,442   261,573   3,779   1,317,119   21,318   11,324   279 
 01/12   22,522,749   375,124   250,249   3,500   1,790,345   22,660   17,217   294 
 01/05   20,732,404   352,464   233,032   3,206   1,499,561   18,435   14,655   233 
 12/29/20   19,232,843   334,029   218,377   2,973   1,258,540   15,460   12,493   186 
 12/22   17,974,303   318,569   205,884   2,787   1,656,411   18,537   16,472   239 
 12/15   16,317,892   300,032   189,412   2,548   1,494,763   17,247   18,825   229 
 12/08   14,823,129   282,785   170,587   2,319   1,375,502   15,483   18,418   175 
 12/01   13,447,627   267,302   152,169   2,144   1,114,175   10,286   15,942   121 
 11/24   12,333,452   257,016   136,227   2,023   1,197,199   10,784   14,130   106 
 11/17   11,136,253   246,232   122,097   1,917   1,099,790   8,501   11,115   65 
 11/10   10,036,463   237,731   110,982   1,852   767,645   6,838   8,868   68 
 11/03   9,268,818   230,893   102,114   1,784   588,207   5,809   5,936   35 
 10/27   8,680,611   225,084   96,178   1,749   492,026   5,585   5,238   (10) 
John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

Books by John Robison:

The Slot Expert's Guide to Playing Slots