The real odds when you’re dealt four cards of a royal flush
By Jerry “Stickman” Stich
It seems players are always looking for proof that the game they love to play is rigged. In an effort to dispel the paranoia, let’s take a look at the real odds of completing several hands.
Any video poker player who has put in time playing the game has been dealt four cards of a royal flush. When this happens, the player will carefully select each of the four cards to be saved—sometimes even allowing the fifth card of a flush to be discarded. Then, while holding their breath, the “deal” button is pressed.
Most often, disappointment is the result. No royal flush, no flush, and no straight. Usually not even a high pair.
This same scenario can happen over and over—five times, 10 times, 20 times or more. It’s enough to make a player think the video poker game must be rigged! Certainly, one time out of 20 should produce a royal flush, right? After all, it’s rare to be dealt four of a royal flush.
I’ve heard this complaint from many people, some of whom I consider friends and savvy players. It seems players are always looking for proof that the game they love to play is rigged. In an effort to dispel the paranoia, let’s take a look at the real odds of completing several hands.
Let’s start with the four cards of a royal flush. Yes, it is rare to be dealt four of a royal. The odds of this happening are about 1 in 2,765, or roughly once every three to six hours of play, depending on your speed. After your initial five cards are dealt, there are 47 cards left in the deck. Only one of these remaining cards will complete the royal, so you have a 1 in 47 chance of completing the jackpot hand.
Keep in mind, this is an average over time. It could take 200 or even 300 times being dealt four of a royal before you complete one. Then again, it could also happen immediately, or even two or three times in a row, but on average you will complete a royal flush from four cards of a royal once every 47 times. It takes a player playing 600 hands per hour almost 217 hours (on average) to play enough hands to complete one four card royal.
Of course, not every failure to produce a royal flush ends with zero money won. There is a 2 in 47 chance of a straight flush if the ace is the only missing card of the royal, and a 1 in 47 chance of a straight flush if the ace is one of the four cards of a royal.
In the remaining 47 cards, there are eight or nine cards of the same suit—depending on whether a card of the same suit was discarded in pursuit of the royal—that will complete a flush. It’s about a 1 in 5 chance if a like suit was not discarded, and just under a 1 in 6 chance if the same suit was discarded.
Assuming the fifth card of a straight was not discarded in the attempt to produce a royal, there is about a 1 in 8 shot of completing a straight. If it was discarded, there is still about a 1 in 9 shot of scoring a straight.
There is also the possibility of snagging a high pair when saving four of a royal. If the 10 is not one of the saved cards, there are 12 cards in the 47 that will match one of the saved cards. This gives you a little better than a 1 in 4 shot at getting your bet back. If the 10 was saved, there are only nine possibly high pair matches remaining—meaning you’ve got a little worse than 1 in 5 odds of returning your bet.
That’s how four of a royal flush plays out. How about the odds of completing other four card hands—such as four of a straight flush, four of a flush, four of a straight, or two pair making a full house?
The odds of completing a straight flush when dealt four are the same as completing a straight flush when dealt four of a royal. If the four cards dealt are all together with no gaps (this is called an open straight or an open straight flush), the odds of completing the straight flush are 2 in 47. If there is a gap, the odds are 1 in 47.
When it comes to four of a flush, there are nine cards that will complete a flush in the 47 remaining cards—about a 1 in 5 chance of snagging a flush.
Like the straight flush, the odds of drawing the correct card to complete a straight depends on whether the cards are together (open) or have a gap (inside). If it’s four cards of an open straight, there are eight cards out of 47 that complete the straight, for just under a 1 in 6 chance. When there is a gap (or the ace – either high or low – is one of the cards), the odds of completing the hand are cut in half, as there are only four cards that will do the trick. There is a 1 in almost 12 chance of completing the hand in this case.
The last four-card dealt hand I will review is two pair. The best possible outcome when holding two pair is a full house. Since there are two more of each of the two pair’s rank in the remaining 47 cards, there is a 1 in almost 12 chance of completing the full house – the same as completing four of an inside straight.
While being dealt four of any potential high paying hand is an exciting proposition, actually completing the hand is still a far from common event. Don’t let your hopes and expectations blind you to the reality of the math. The math will win out in the end.
Video Poker Strategy – How Would You Play This Hand?
This month’s article showed the odds of completing various combinations of four card hands. Now, use this information to select the proper play. Playing full pay Jacks or Better with 5 credits played, you are dealt the following hand:
As Jh Qd 9c Th
How would you play it?
You could save the Ace, Jack, Queen and Ten going for a high pair, two pair, three of a kind, or straight. You could save the Ace, Jack and Queen going for a high pair, two pair, three of a kind, or straight. You could hold the Queen and Jack going for all the previous as well as a possible four of a kind. You could hold the Jack and Ten going for a high pair, two pair, three of a kind, straight, flush, four of a kind, or a royal flush. You could hold the lone Ace, Jack or Queen and go for all of the previous as well as a straight flush.
The best hold however is the Jack, Queen, Nine and Ten as this is an open straight.
Returns for each mentioned save are as follows.
Jack, Queen, Nine and Ten – 4.04 credits
Ace, Jack, Queen and Ten – 2.66 credits – nearly half the optimum hold
Jack and Queen – 2.41 credits
Jack and Ten – 2.33 credits
Ace and Queen – or – Ace and Jack – 2.32 credits
Ace only – 2.24 credits
Queen only – 2.23 credits
Ace, Jack, and Queen – 2.21 credits
Jack only – 2.20 credits
Is that how you would have played it?