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# Chances of hitting the royal flush

1 July 2011

Video poker players hope the next hand will give them a royal flush because the payoff for a five-coin bet is an immediate 4,000-coin payoff. A reader sent me a question about the player’s chances of hitting a royal if a player held three cards to the royal and drew two cards, or held two cards to the royal and drew three cards. I decided to answer her question in my column this month.

First off, have you wondered what your chances are of being dealt a three- or four-card royal on the initial deal? How about a five-card royal flush on the initial deal? That latter would be sweet and as you will see shortly, the odds are long but not impossible.

Table 2 (page 164) in Dan Paymar’s book, Video Poker-Optimum Play, contains a summary of Pre-Draw Hands for 9/6 Jacks-or-Better, and I’ve summarized the results in the table below for all possible hands that you could be dealt that contain from 0 to 5 royal flush cards.

 Pre-Draw Hand Chance in % Which Equals: RF5 0.00015 1 in 649,740 RF4 0.036 1 in 2,777 RF3 1.09 1 in 92 RF2 7.7 1 in 13 RF1 16.7 1 in 6

As you can see in the table, your chance of being dealt a five-card royal flush on the initial deal (RF5) is a minuscule 0.00015 percent (1 in 649,740). You might think that with those long odds, you’ll never be lucky enough to be dealt a royal flush. However, never say never. Two weeks ago, I was dealt a 10-J-Ace-Q-K in spades on the deal for an instant royal flush. It happened quickly because I was playing fast (I had the speed of the cards being dealt on the screen on the fast setting). I knew something good happened when the machine locked up and music started playing. My second dealt royal flush was even more memorable. I was showing my father-in-law how to play a Triple Play game when I was dealt a royal flush on the bottom hand (and of course, I automatically wound up with a royal on the second and third hands). I’m not writing about these two royal flushes to brag, but to make a point. Even though the odds are long, getting a royal on the initial draw could happen to anyone at anytime.

I’ll bet you didn’t know that once in every 2,777 hands you’ll experience the thrill of being dealt four cards to the royal. Moreover, a little less than every 100 dealt hands (92 to be exact), you’ll have a three-card royal flush. But let’s get to the question raised by the reader: namely, what are the chances of hitting a royal after being dealt RF4, RF3, RF2, or RF1?

Suppose you are dealt these five-cards: A-K-Q-J-5, where the first four cards are spades. Your heart starts pumping when you realize you have a four-card royal flush. You hold the four-card royal, say a quick prayer, and hope that you get the 10 of spades on the draw for the royal flush. So, what are your chances of getting the card you need?

Video poker is played with a standard deck of 52 cards (assuming you aren’t playing a joker-wild game), and five of the 52 cards were used on the initial draw. When you hit the draw button, you will get one card from the 47 unplayed cards and, of course, you are hoping that card is the 10 of the spades. In the pile of 47 unplayed cards, there is only one 10 of spades. If you draw that card, and only that card, you’ll get a royal flush. If, instead, you draw any of the other 46 unplayed cards, you will not get the royal flush. Therefore, the chance of hitting the royal flush when you hold a four-card royal flush and draw one card, is 1 in 47. This means for every 47 times you hold four to the royal, on average you will hit the royal once, and not hit the royal the other 46 times. But notice I said "average," meaning you could hit more than one royal or even no royals after every cycle of 47 draws.

While I’m on this subject of four to the royal, let me bring up a strategy point. Last night while playing video poker, I was dealt this hand: 10-J-A-K-Q, where every card was a diamond except the Ace of hearts. In other words, I was dealt a paying five-card straight that also contained the four-card royal flush in diamonds. If I held all five cards, I would have been paid \$20 instantly for the five-card straight (dollar denomination machine). Would you have taken the sure \$20 payoff and held the straight? If you answered yes, you would have a made a big mistake. When you are playing Jacks-or-better and you are dealt this hand, the play that has the higher expected value (EV) is to hold the four-card royal flush, discard the fifth card, and draw one card and hope you get the card you need for the royal flush. Sure, most of time you will come up empty handed on the draw, but the payoff on the one time you get the card you need for the royal will more than make up for the times that you wound up with nada. (When I hit the draw button, the Ace of hearts disappeared and up popped the Ace of diamonds, a 1 in 47 shot, for a \$4K royal flush, proving the point that sooner or later you will get the card you need when you draw one card to a four-card royal flush).

Now let’s suppose you are dealt this hand: A-K-Q-6-3, where the first three cards are spades and the other two cards are a different suit. Now, what are your chances of drawing the two cards you need for the royal flush when you hold three to the royal? Mathematically, there are 1,081 different hands that you can make when you hold A-K-Q in spades and hit the draw button. Only one of those hands will result in a royal flush; therefore, the chance of hitting a royal when you hold three cards to the royal is 1 in 1,081 (0.09 percent).

The following table summarizes the chances of hitting a royal flush depending on how many royal flush cards you hold in the initial hand. In the case of holding two cards to the royal (RF2) and drawing, you’ll hit the royal once in every 16,215 tries. The chance of getting a royal when you hold only 1 high card (RF1) or no high cards (RF0) are slim, but you might get lucky and beat those long odds. My father-in-law was once dealt five garbage cards, properly discarded all of them, and was dealt a royal flush on the draw -- a 1 in 383,484 event.

 # RF Cards in Initial Five-Card Hand Chance of Hitting the Royal Flush RF0 1 in 383,484 RF1 1 in 178,365 RF2 1 in 16,215 RF3 1 in 1,081 RF4 1 in 47

The term “royal flush cycles” in video poker often confuses players and I’ll cover that important point in a future article.

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Henry Tamburin

Henry Tamburin is the author of the best-selling book, Blackjack: Take The Money and Run, editor of the Blackjack Insider e-Newsletter, and Lead Instructor for the Golden Touch Blackjack course. For a free 3-month subscription to his blackjack newsletter with full membership privileges, visit www.bjinsider.com/free. For details on the Golden Touch Blackjack course visit www.goldentouchblackjack.com or call 866/WIN-BJ21. For a free copy of his casino gambling catalog featuring over 50 products call 888/353-3234 or visit the Internet store at www.smartgaming.com.

#### Henry Tamburin Websites:

www.smartgaming.com

#### Books by Henry Tamburin:

Henry Tamburin
Henry Tamburin is the author of the best-selling book, Blackjack: Take The Money and Run, editor of the Blackjack Insider e-Newsletter, and Lead Instructor for the Golden Touch Blackjack course. For a free 3-month subscription to his blackjack newsletter with full membership privileges, visit www.bjinsider.com/free. For details on the Golden Touch Blackjack course visit www.goldentouchblackjack.com or call 866/WIN-BJ21. For a free copy of his casino gambling catalog featuring over 50 products call 888/353-3234 or visit the Internet store at www.smartgaming.com.

#### Henry Tamburin Websites:

www.smartgaming.com