Looking at Expected Value: The Dreaded Dealt King-High Straight Flush

^{Frugal Video Poker, with modifications.}
Suppose you're playing non-progressive video poker and you get dealt the hand in the above image. It's a straight flush, but that straight flush is also one card off from a royal flush and the jackpot of 800 times your bet that comes with it. There are only two viable plays here: keep that straight flush, or toss the 9 and hope you grab the suited Ace for the royal flush. So what's the best play? That turns out to be dependent on what game you're playing. Today we'll look at a few of these cases.

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Ordinary 52-card Deck Games

In most forms of video poker without wild cards, that dealt straight flush is worth 50x your bet. A few paytables lower the straight flush to 40x, while others increase the straight flush award. Let's look at how this plays out on full-pay Jacks or Better and Double Double Bonus games (9/6 for both, but you can also count the rare 10/6 DDB version too).

If you choose to toss the 9, the same set of possible outcomes and payoffs are available in both games:

• 1 card is the suited Ace that will complete the royal flush (pays 800x).
• 7 cards, the 2-8 of the matching suit, will give you a flush (6x).
• 6 cards, the other Aces and 9s, will give you a straight (4x).
• 9 cards, the remaining Jacks, Queens, and Kings, will give you a pair of Jacks or Better (1x).
• The remaining 24 cards will result in a losing hand (0x).

The expected value (EV for short) of tossing the 9 then, is calculated as the average of all the possible outcomes.

[(1 * 800) + (7 * 6) + (6 * 4) + (9 * 1) + (24 * 0)]/47 = 875/47 = 18.6170...

Thus, tossing the 9 has an expected value of about 18.617x your bet. The suited Ace alone contributes just over 17x to the EV. So no matter whether the straight flush is paying the normal 50x, the stingy 40x, or something more generous, it's a very bad idea to break up a dealt straight flush to go for the royal in these games.

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Deuces Wild? Things Change.

In Deuces Wild games, straight flushes pay a lot less, even if the straight flush is made naturally without any 2s. Depending on the variant and paytable, a straight flush pays 6-13 times your bet. Lower-ranked hands also pay less accordingly, e.g. flushes are typically worth 2-3x your bet (plus, the lowest-paying hand is 3-of-a-kind).

One could choose a particular Deuces Wild variant and paytable and run the same EV analysis as we did before on a dealt King-high straight flush. I will leave this as an exercise to readers who wish to do such an analysis. For the purposes of making a decision on whether or not to go for the royal, a full analysis is unnecessary. Just like before, the suited Ace by itself will make a natural royal flush paying 800x, and that alone contributes just over 17x to the EV of tossing the 9. So without even considering other potential winning hands (e.g. wild royal flushes for drawing any 2), we see that there will always be greater EV in going for the royal than keeping the dealt King-high straight flush (17 > 13, and if a straight flush is paying 13x it's almost always on a very bad paytable too).

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The Takeaway

Being dealt a King-high straight flush in video poker isn't the most wonderful thing in the world, but the correct course of action to maximize a player's EV and returns will depend on the game being played. In non-wild card games that start paying for a pair of Jacks or Better, you should keep that dealt straight flush, not get too greedy, and take the nice payout. In Deuces Wild games, you should always be brave and go for the natural royal.

Until next time, good luck!