What Are the Odds of a Royal Flush Holding Just One Card?

6 November 2012

By Jerry Stickman

Mr. Stickman,

Yesterday I was at my favorite Midwestern casino playing one of my favorite video poker games. I was dealt a hand where I saved an ace of hearts. I play fairly fast and once I hit the draw button I start hitting the deal max button rapidly to get the next hand dealt as soon as possible. What a surprise! The game locked up after showing the prettiest sight on a video poker game a royal flush in hearts.

My question is what are the odds of completing a royal flush when saving just one card?

Yours truly,
One Lucky Gal


Hi One Lucky Gal!

Congratulations on your good fortune. Depending on the game you are playing and the pay table, a royal flush only happens about every 35,000 to 45,000 hands.

The odds of being dealt a royal flush are 1 in 649,740 which is significantly higher than the 35,000 to 45,000 mentioned above since these take into account saving the proper cards and redrawing.

If anyone is interested in how to calculate the odds of completing a royal flush with one card saved, here is how it works. You are dealt five cards out of a 52 card deck and you save one of them (the Ace of Hearts).

There are now four cards in the remaining 47 cards that will help complete the royal flush (King, Queen, Jack, and Ten of Hearts). Once you get another card of the royal (assume the King of Hearts), there are now three cards in the 46 cards left in the deck that will help complete the royal (Queen, Jack, and Ten of Hearts). The process continues with two out of 45 cards remaining, and one out of 44 cards remaining. These are put into an equation like this: 47/4 * 46/3 * 45/2 * 44/1. Then simply calculate the answer. The odds of completing a royal after saving one card are just 1 in 178,365.

One Lucky Gal, as you can see (and probably suspected) you certainly beat the odds this time.

May all your wins be swift and large and all your losses slow and tiny.

Jerry “Stickman”

Jerry “Stickman” is an expert in craps, blackjack, video poker and advantage slot machine play. You can contact Jerry “Stickman” at stickmanGTC@aol.com