Ask the Slot Expert: Six royals in a cycle

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.

Click here for the latest Covid data.