Getting Your Share of Royal Flushes

“ Who is gonna make it? We’ll find out in the long run.
We can go the distance if our love is a strong one.” — Don Henley and Glenn Frey, The Eagles

Your success as a Video Poker Player is largely determined by the number of Royal Flushes (excluding Wild Royals) that you get compared to the number that the odds predict. Assuming that you are playing a game with an expected return (including comps) of at least 100.5%, you will come out ahead if you get about 75% or more of the expected number of Royal Flushes. This is because Royal Flushes contribute about 2% of winnings on the average. So if you get 75% of your share, the contribution will be 1.5%, half a percent below average. This reduces your total return to breakeven.

So the questions are, how many Royal Flushes can you expect to get?, and what are your chances of coming out ahead? We’ll answer these questions with tables and graphs.

We start with the probability of getting a Royal Flush in a single hand. For most games, this probability is about 1 in 40,000, and that is the number we use for these calculations.

Intuitively we might expect to average one Royal Flush for each 40,000 hands we play, and this intuition is correct. Accordingly we will use the term Royal Flush Cycle to mean 40,000 hands. So 4,000 hands is a tenth of a Cycle, and a million hands is 25 Cycles.

What we want to know is, given that we have played a certain number of hands, what are the probabilities of getting no Royal?, 1 Royal?, 2 Royals?, etc. The graphs that follow illustrate these probabilities.

Many of the questions about probabilities that we want to answer involve queries like “given that we have played a certain number of hands, what is the probability that we get five or fewer Royals?” Table 1. at the end of this article gives the answers to these questions, and we will use it in discussing the graphs.

Why don’t we also consider the number of quads that you get in estimating your return? Because by the time you have played enough hands to make your Royal Flush expectations stable, the expected variation in quads is very small and has a negligible effect.

First, suppose we have played 4,000 hands or 1/10 of a Royal Cycle. At 500 hands per hour, this is about 8 hours of play. For our purposes, let’s agree to call this a day’s play. The following graph shows the probabilities

This graph tells us that on any given day the probability of getting one or more Royal Flushes is slightly less than 10%. More than 90% of the time, we won’t get a Royal. One day in about 200 we will get two Royals, and one day in 5,000 we will get three.

Now suppose we play 20,000 hands, or half a Royal Cycle, about five days play, a typical Las Vegas visit for many out-of-towners:

More than 60% of the time, we won’t get a Royal. 30% of the time we will get one. More than 7% of the time we’ll get two. One trip in about 80 we will get three. One trip in 5,000 is more than a lifetime, so we can’t expect to ever get five.

Next suppose we play 100,000 hands. For those who visit Las Vegas five times a year for five days at a time, this is a year’s play.

So we are going to get shut out once every 12 years. The probability of getting five or more royals in a year is about 10%, so one year in 10 will be thrilling. Notice that the graph is beginning to look like a bell-shaped curve.

After ten years at this rate, we will have played about a million hands:

For those of you still chasing your first Royal, (including, as of this writing, my wife Meg) take heart. The probability that you get less than seven royals in a million hands is less than 1 in a hundred thousand. The probability that you will get 25 or more is close to 53%.

The shape now is very much a bell, and illustrates a famous principle in statistics called the Central Limit Theorem (sounds heavy duty, doesn’t it?). It states that with sufficient repetition, sample averages from almost every probability distribution are distributed in a bell-shaped curve, centered about the expected average, with the shape getting narrower as the sample size gets larger. Translated into Video Poker terms, the more hands you play, the more likely it is that the number of Royals you get is close to the number of Royal Cycles you have played.

Finally, suppose we have played 4 Million hands, about 40 years using the same play rate, a lifetime of play for many of us:


Now the probability that you get less than 60 royals is less than one in a hundred thousand, and the probability that you get at least 100 exceeds 51%.

Finally, we want to examine the question, “What is the probability that I will come out ahead?”. The answer depends very much on how many hands you play. Using the rule of thumb cited above that you need to get at least 75% of the expected number of Royals, we can construct the answer in the form of a table:

Is this is a glass half-empty or a glass half-full? After 160,000 hands playing a game with an expected return of 100.5% almost 25% of players will be behind. On the other hand, almost everyone, including most of the original losers, will be ahead when they get to 4,000,000 hands. Conclusion — by playing more and playing faster you can significantly reduce the probability of being behind — yet another reason to practice to improve your playing speed.

Cap Richards, CVPN (Certified Video Poker Nut) is the creator of Video Poker Wizard Coach, Training and Practice Software at www.VideoPokerWizard.com (“Speed Training for Video Poker Players”). He is a Principal in an Investment Advisory Firm, President of a Small Internet Services Provider, and formerly President of a company that provides software design and development tools to large corporations and government agencies. Cap and his wife Meg live near Nashville, and visit Las Vegas not nearly as often as they would like. Contact Cap at CapRichards@VideoPokerWizard.com.