# Inside video poker's crown jewel the Royal Flush

June 20, 2017 3:00 AM
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I’ve often felt I look at situations very differently than the average person. This doesn’t make me better or worse, just different.

This past week one of those situations happened. A friend of mine posted on Facebook the picture of his final video poker hand – a Royal Flush. While everyone else was congratulating him, I thought – “I wonder how many cards he held.” I was, of course, happy for my friend. He won \$1,000, and while this won’t change his life much, there is no such thing as a bad Royal Flush.

But my mind always works on the numbers and the strategy. Anytime I hear about a new game, my mind immediately starts thinking about what the strategy looks like and where the house advantage is and where the player can equalize it out. Others will simply think about whether the game is fun, which is a critical component of a game’s potential success. But after many years of always looking at the math, it is where I look first.

The screenshot he put up on Facebook showed he held a lone King. That, of course, led to me wonder what else he had and did he play the hand properly. Without asking him directly, I’ll never know this last piece. Playing high cards seems so simple. If you hold one, then certainly you would hold two if you had them. And if two is worth playing, the three must be even better. Well, not exactly.

All these hands are pretty bad. They are near the bottom of our strategy table. Ironically, what gives them a fair amount of their expected value is the rare shot of hitting the Royal. But, this means if you hold two or more unsuited cards, the chance for a Royal goes out the window. The second winning hand that gives these hands some power is a Straight. But, if you hold an Ace out of the high cards, it immediately becomes essentially a closed ended straight with only one way to hit it. So, we have to be more careful than one might think on these hands.

While three High Cards outranks two High Cards and one High Card on our strategy table, it does not outrank most of our 2-Card Royals. This means if two out of our three High Cards are of the same suit, we keep only those two. From this, we learn that three High Cards must be of three different suits. Next, only JQK is worth keeping as three High Cards. Throw in an Ace and the expected value goes down and now the two High Cards has a higher expected value. So, if you have JQA offsuit, you hold the JQ only.

The rule for two High Cards vs. one is much easier. In this case two is better than one and you keep both High Cards no matter what they are. Obviously, if they are suited, it counts as a 2-Card Royal and the expected value will be that much higher.

So, back to my friend’s hand. Holding a single high card, there are just over 178,000 possible draws. Only one will result in a Royal Flush. In the case of the single High Card, the Royal does not contribute a huge amount to the expected value. Even 800 units divided by 178,000 draws simply doesn’t add up to much. But, with two High Cards (suited), it adds about 0.05 to our expected value, which explains why the 2-Card Royals outrank the 3 High Cards.

Drawing four to a single High Card is not the most likely way to hit a Royal. Believe it or not, the most common way is from a 3-Card Royal. Obviously, once dealt a 4-Card Royal, you are more likely to get a Royal than from a 3-Card Royal. But, when we take into account the frequency of being dealt a 3-Card Royal vs. a 4-Card Royal, we find about 40% of our Royals will happen this way. Only about 3% will come from a single High Card.

As bad as the one High Card hand is, we have to remember it is the second most common dealt hand (behind Low Pair). As such, we must resist any temptation to play the hand in a variety of other ways that might seem reasonable before we do the math. We do not hold 3-Card Straights or 3-Card Flushes. These hands simply do not have enough upside to ever be held.

While we don’t count on hitting a Royal in this manner, when they do happen, it sure is a good feeling.