Great Expectations – Royal Flush

Editorial Staff | Posted on July 25, 2010 |

When can you expect a royal flush, and how do you know if a machine isn’t giving you a fair shot?

By Henry Tamburin

There is a mathematical formula that you can use to calculate the probability of hitting any number of royal flushes in any given number of cycles. The formula is called the Poisson Distribution, and it’s named after the French mathematician Simèon Poisson—who developed the formula in the 19^th century to calculate the probability of rare events.

A Strictly Slots reader sent me this question:

“Please explain what is meant by a ‘royal flush cycle.’ I thought video poker machines were random, and you can’t predict when you are going to be dealt a royal?”

A royal flush cycle is the mathematically calculated average number of hands it takes to hit a royal flush, when you use perfect strategy. The number of hands in a royal flush cycle varies slightly from game to game. On a jacks or better machine, the royal flush cycle is 40,391 hands, whereas on a full pay deuces wild game it’s 45,282 hands (See Table 1).

Why is there a difference in the number of hands? Because in some games, the playing strategy calls for holding more two or three-card royal flushes—which means you can receive more royals.

Table 1

Royal Flush Cycles

Most players expect to hit one royal flush after playing roughly 40,000 hands. But this is not necessarily the case. The math says on average you will hit a royal flush once in every 40,000 hands, which means for a whole bunch of sets of 40,000 hands, you’ll average one royal.

In other words, in any given one set of 40,000 hands, you could wind up with more than royal or, heaven forbid, possibly no royals. (Would you care to guess what the chances of the latter catastrophe occurring? Keep reading for the answer.)

There is a mathematical formula that you can use to calculate the probability of hitting any number of royal flushes in any given number of cycles. The formula is called the Poisson Distribution,  and it’s named after the French mathematician Simèon Poisson—who developed the formula in the 19^th century to calculate the probability of rare events.

Don’t worry. You don’t have to pull out your calculator, because my friend and fellow video poker author Dan Paymar (author of Video Poker Optimum Play) has done the work for us. His calculations yield the following results for one cycle of 40,000 hands (jacks or better):

Table 2

Probability of Hitting a

Royal in One Cycle

Wow! If you look at the data in Table 2, it says that you have the same 36.8% chance of getting one royal or no royals after playing one cycle of 40,000 hands. Since the royal flush contributes 1.98% toward the overall 99.5% ER for jacks-or-better, your return between royals is only 97.5% (meaning that your bankroll will more than likely head south from one royal flush to the next one).

This, dear readers, is why you must have enough bankroll to play video poker…to cover those times when you play many hands without hitting a royal.

Have I ever played one cycle and not hit a royal? You betcha, and it was painful. I keep records of all my playing sessions, and the worst streak I’ve ever had was about 85,000 hands without a royal flush. Even though that was painful, I have friends who play video poker that have gone way more than 85,000 hands between royals (ouch!).

But let’s be optimistic and look at the other side of the curve. I just came back from a trip to Las Vegas where I played roughly 40,000 hands of jacks or better (one cycle) and I hit three royal flushes. Was I lucky? According to Table 2, the chance of hitting three royals in one cycle of jacks or better is a paltry 6.1%. So yes, I would consider myself very lucky to have hit three royal flushes during that trip.

The data in the above table also leads to this conclusion: You have about a 63% chance of hitting one or more royal flushes in one cycle, and only a 36.8% chance of hitting no royals (does that make you feel any better?).

Do you think it’s impossible to play 200,000 hands of jacks or better (five cycles)—which many video poker aficionados consider to be the “long term”—without hitting one measly royal flush? I hate to be the bearer of bad news, but according to the data in Table 3, you have a 1 chance in 140 (0.7%) of not hitting a royal flush even after playing 200,000 hands (or roughly 400 hours of play).

Table 3

Probability of No Royals after X Cycles

The percentages in Table 3 are scary. You have a 5% chance of getting no royals after 120,000 hands (3 cycles), and a 1.8% chance after 160,000 hands (5 cycles). Even though the chances of winding up without a single royal flush are slim, if it were to occur, it could be financially catastrophic (especially if you don’t have enough bankroll to weather this remote, but still possible, outcome).

The reality for video poker players is this: In any one royal flush cycle of roughly 40,000 hands, there are no guarantees that you will hit exactly one royal flush. If you are lucky, you could get more than one royal flush, and if unlucky, you could wind up without any royals. This is why having a sufficient bankroll is very important.

Tamburin’s Tip of The Month

A general rule when you play jacks or better, or bonus poker, is to hold a low pair over any four-card straight (with no gaps). However, there is one specific four-card straight that you should hold over a low pair—and it’s 10-J-Q-K.

Even though all four-card straights with no gaps have the same chance of becoming a paying straight, the four-card straight 10-J-Q-K has three high cards, which improves your chance of getting a high pair on the draw. For this reason, holding the 10-J-Q-K straight is a better play over holding a low pair. Therefore, you should hold the four-card straight 10-J-Q-K over a low pair, and the latter is held over any other four-card straight with no gaps.

Henry Tamburin is a blackjack and video poker expert. He hosts the smartgaming.com website and is the editor of the Blackjack Insider newsletter (for a free three-month subscription, visit www.bjinsider.com/free). For a FREE copy of his Casino Gambling Catalog, which contains training products for video poker players, call toll free 1-888-353-3234, or visit the web store at smartgaming.com.

When can you expect a royal flush.