“Why Didn’t I Hit A Royal Flush?”
Understanding the royal flush cycle
By Henry Tamburin
I received the following email me from a reader that I decided to make the topic of my column this month.
Following your advice in a column I read in Strictly Slots, I’ve been playing 9/6 Jacks or Better for over a year but I still haven’t hit a royal flush. You wrote that a royal flush occurs once in every 40,000 hands and I know I’ve played more than 40,000 hands this past year? What’s going on here?
Background: If you are not familiar with the term “royal flush cycle” let me explain. A royal flush cycle is the mathematically calculated average number of hands it takes to hit a royal flush with perfect playing strategy. The number of hands in a royal flush cycle varies slightly from one game to another. For example, one royal flush cycle for Jacks or Better is 40,391 hands, whereas for NSU Deuces Wild, it’s 43,456 hands. The reason for the slight differences in the number of hands per cycle is because in some games the playing strategy calls for holding more two- and three-card royal flushes than other games. However, for the purpose of this article, I will use an average of 40,000 hands per one royal flush cycle.
Answer to the question: Every time I write about a royal flush cycle, I include the words “on average.” What that means is that you should not expect to hit exactly one royal flush after playing exactly 40,000 hands. The math says that on average you will hit a royal flush in every 40,000 hands. This means for many sessions of 40,000 hands, you will average one royal flush. So it’s possible to play exactly 40,000 hands and not get any royal flushes, or perhaps one, or even two or more, royal flushes.
Take a gander at the table below, which shows the probability a royal flush in one cycle. (From the book, Video Poker Optimum Play by Dan Paymar.)
PROBABILITY OF HITTING
A ROYAL IN ONE CYCLE
|
# OF ROYALS |
PROBABILITY |
|
None |
36.8% |
|
1 |
36.8% |
|
2 |
18.4% |
|
3 |
6.1% |
|
4 |
1.5% |
|
5 |
0.3% |
|
6 |
0.1% |
What the data in the table tells us is that you have a 36.8% chance not getting a royal after one cycle of 40,000 hands, the same 36/8% chance of hitting one royal, 18.4% chance of hitting two royals, and so on. So to answer the reader’s question, yes, it’s possible that you could play 40,000 hands and not hit one measly royal flush even though you are playing a high return video poker game with perfect strategy. (This is why I harp about the importance of having enough bankroll when you play video poker in the event you wind up with no royals after one cycle.)
If you think it’s painful playing one cycle without a royal flush, how about going through three cycles without one (about 120,000 hands). That is, in fact, what recently happened to my wife Linda, even though she is an expert player, who predominately plays 9/6 Jacks or Better Super Times Pay with a 99.82% return. The probability of getting not one royal after three cycles is only 5% but Linda still managed to experience this stressful situation. Thankfully, she had enough bankroll to weather the painful downswing in her bankroll during this royal flush drought until she finally hit not one, but two royal flushes within one hour of each other. In fact, on this particular trip to Las Vegas Linda played roughly 40,000 hands of video poker (one cycle) and wound up hitting a total of three royal flushes. Was she lucky? The math says (see table) you have only a 6.1% probability of hitting three royals in one cycle; therefore, it’s fair to say that she was very lucky to have hit three royals on this trip. (The point of mentioning this real-world example is not to gloat but rather to show you the two extremes that could happen to you when you play video poker even with expert strategy… from no royals in three cycles to three royals in one cycle.)
Here’s the last bit of information to ponder: the data in the table says that you have a 63% chance of hitting one or more royal flushes in one cycle and only a 37% chance of getting no royals. (Does this make you feel any better?).
Tamburin’s Tip of the Month
Take a look at the following hand and tell me how you would play it (assume no-wild card game).
6♥ 8♣ 2♥ 9♠ 4♥
At first glance, there doesn’t appear to be any cards worth holding, and if you are like most players, you’d discard all five cards and draw five new ones. But that would be a mistake.
If you look carefully at the hand, it contains the three-card straight flush with two gaps (2-4-6). Players often miss this hold because it’s not always obvious. Holding the 2-4-6 doesn’t seem like such a great play but it is still better than drawing five new cards. For example, if you were playing 9/6 Jacks or Better, the Expected Value for holding the suited 2-4-6 is 2.21555 whereas it’s only 1.80795 for drawing a new hand. (For a dollar player, it would cost you 41 cents every time you drew five new cards instead of holding the suited 2-4-6, and over time, this adds up). So the next time you glance at the cards in your hand and think you have nothing worth holding, take another look to be sure you don’t have a three-card straight flush with two gaps.
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Henry Tamburin is a blackjack and video poker expert. He is the host of the smartgaming.com website and the editor of the Blackjack Insider newsletter (for a free three-month subscription, visit www.bjinsider.com/freetrial). For a free copy of his Casino Gambling Catalog, which contains books, strategy cards, and software for casino players, call toll free 1-888-353-3234, or visit the web store at smartgaming.com.
