Get in step with
royal flush cycles

Mar 8, 2005 8:41 AM

With the advent of games like Double Double Bonus poker, some pressure has been taken off of the royal flush as the sole hand to be considered a jackpot. Quad aces with a 2, 3 or 4 kicker can pay as much as half a royal while occurring far more frequently.

This leaves the royal flush as the critical hand for most versions of video poker. Ignoring progressives for a second, royals account for about 2 percent of the payback of most games. Given this, even a full-pay jacks or better machine paying 99.5 percent is reduced to 97.5 percent while you’re not hitting that royal. This can make for a tough session without one. In the long run, to win at a game like jacks or better, you have to get your share of royals. What does this really mean?

While using expert strategy when playing a full-pay jacks or better machine, a royal flush should show up about once in 40,400 hands. Playing at 600 hands per hour, this means you should hit a royal every 67-68 hours of play. So, if you go to Las Vegas two weeks a year and play five hours a day, you’re about guaranteed to get a royal, right? If only it were that simple.

If an event is supposed to occur once every 68 hours, how likely is it to occur only 10 hours apart, or perhaps as spread out as 200 hours? Or should we expect most occurrences to be very close to 68 hours apart?

I suppose I could try and demonstrate the answer through anecdotal evidence. When I was younger, I used to go out to Las Vegas for several weeks a year and play video poker for two to three hours per night. It took five or six years of playing before I hit my first royal flush.

A month later, I hit my second. Later that week, I hit a third. Aside from hitting one on a 100-play machine (which I don’t really consider hitting a royal), that was the last one I hit, roughly seven years ago.

Admittedly, I don’t go out to Las Vegas as often since my dad passed away, but there has been far more than 68 hours logged in those seven years! On one of my earliest trips to Las Vegas, a woman behind me hit a royal for a nickel and said it was the first time she had ever played video poker and had only been playing for an hour!

Anecdotal evidence tends to be amusing, but hardly reliable. Instead, I prefer to rely on mathematical models and computer programs to analyze these types of situations. In this case, I decided on the latter. I set up a program to simulate an event that would occur once every 40,400 hands — our royal. I then played out 1 billion hands and kept track of the interval between royal flushes. I found the results to be fascinating.

Over the 1 billion hands, we drew 24,752 royals, or one every 40,400.78 hands. About 1.1 percent of these occurred in the 67- or 68-hour mark after the previous royal. About 7 percent of the royals occurred between the 62- and 73-hour mark, or within five hours of the average.

In fact, it was more common that a royal occurred within the first five hours since the previous royal than in this 11 hour span around average. Nearly a quarter of royals will occur after more than 150 hours, or more than twice average, while about two-thirds of our royals will occur in less than the average span.

So what do we learn from all this? First of all, the word average can be rather misleading. It does tell us that, because it accounts for 2 percent of the payback, we need to see one every 40,400 hands in order to have a good chance of achieving our expected payback of 99.5 percent.

It does not mean that we should expect to see one every 40,400 hands or so. While my own personal experiences probably highlighted some less than common examples, it’s pretty clear to me that I have been dealing with nothing out of the ordinary. It is not that uncommon to hit two royals within a few hours or to go a hundred hours or more without hitting one. I know, because I’ve lived them both. Twice!