How Lucky Was It?

Last week I wrote about completing a royal flush cycle (one or more royals in each suit) within one trip. I was very lucky, to be sure, but just how lucky?

I have a hard time calculating how lucky something was after the fact. You can massage the numbers and come up with all sorts of probabilities. There is not one absolutely correct answer that everybody can agree on.

Let’s start with the 11-day trip. I used all of it, hitting the club royal on my last day there. Most players never have an 11-day out-of-town casino trip in their entire career! I’ve had 14-day trips to Cherokee. I would have been just as delighted to complete the cycle on one of those trips. But I never did.

It’s not the length of the trip, of course, that determines your chances, but rather the number of hands played. Near as I can tell, I played about 62,000 video poker hands. I know how many points I earned during the trip, but some of my play was on slots. I record my points earned each day, but not how that number of points is broken down between slots and video poker.

I know I’m going to be using the word “cycle” in a different way now, but a “royal cycle” is the number of hands, on average, to hit a royal flush in a particular game. In Jacks or Better the number is right around 40,000, and that’s the commonly used number for video poker royal cycles. But NSU has a longer cycle, 43,456 hands, because you play hands differently in this game.

Calling 62,000 hands 1.4 royal cycles is as close as I can get. The 43,456 number is fairly precise (I could tell you that number is 43,456.27, which would be more precise, but hardly more useful), while the 62,000 number is an educated guesstimate. Using the Binomial Theorem, connecting on exactly four royals in 1.4 cycles happens about 4% of the time, and connecting on four or more royals in 1.4 cycles happens about 5.6% of the time.

If I connect on exactly four royals, all four suits will be present only about 9% of the time. Were I fortunate enough to have connected on five-or-more royals (I wish!), it would have been easier to have all four suits represented. Not a lock, of course, but easier.

Now what do we do with the club royal being dealt? The “dealt-ness” of that royal was overkill. I would have also completed the cycle even if I had needed to draw one or more cards to get the club royal. 

But the royal being dealt allows me to jack up my numbers when I tell people how rare this was. (It’s not something I normally do, but I’m discussing it here because there are always “How rare was it anyway?” questions.)

The dealt royal arrives approximately every 650,000 hands. But since at the time it hit, I needed the royal in clubs to complete the cycle, those only come around one-quarter as often — or about every 2,600,000 hands. 

All these things had to happen on the same trip — namely playing 62,000 hands, collecting at least four royals, having every suit being accounted for in those four or more royals, and (this one is optional), one of these had to be dealt. To determine how likely all of this is, you need to multiply all of these probabilities together. I’ll let others do it, because I’m not at all convinced that figuring out how likely something was to happen — after you know it did happen — is a meaningful exercise at all.

There’s more on this trip. The deuces cycle in this game is about 5,356 hands. In 62,000 hands you have 11.6 of these cycles. I collected 12 sets of deuces — which is essentially spot on given the imprecision of the 62,000 number. The thing is, one of those sets of deuces was dealt. 

Being dealt a specific quad happens every 54,167 hands, on average — so in 62,000 hands “it figures” I would have collected one or more. Mathematically, even though 62,000 is larger than 54,167, I was still a slight underdog to hit exactly one set of dealt deuces on the trip, although I was a sizeable favorite to collect one or more.

What this has to do with anything is that on the same trip I was dealt a royal and dealt deuces! (I was also dealt four aces with a deuce, which is another rare event that is called a 5-of-a-kind for $400 in $5 NSU Deuces Wild and it’s the kind of hand that makes you wish you were playing a different game!) Being dealt a royal is rare enough. But also being dealt deuces is even more rare!

It was, to be sure, a trip to remember!

Sometimes casinos restrict players who have too much success. I’m hoping that’s not the case here.