Royals and Flushes

In my August article, Grit Your Teeth and Go For It, I considered the 9/6 Jacks or Better Video Poker situation wherein you are dealt a four-card Royal and a pat Flush. Although giving up the sure winner is tough to do, we saw there that the correct play was to break up the pat Flush and hold the four-card Royal. In fact, the decision wasn't even close. This month I want to consider a similar situation in which the player is dealt a three-card Royal and a four-card Flush. Specifically, I want to look at the weakest three-card Royal one can hold, that being the Ace-Ten with another face card. Why do I call this "weakest?" Because there is only one type of straight one can fill and the ten cannot be used to form a high pair.

For definiteness let us consider the hand AH, QH, TH, 8H, 6C. Obviously we're going to toss the six; what about the eight? The remaining deck contains 47 cards of which 9 are hearts. If we keep the four Flush then we can obtain a flush 9 ways and a high pair six ways (3 Aces and 3 Queens). That's it. The expected return table looks like the following:

Dividing 300 by 47 we obtain the expected return for this four-card hand; it is 6.383. Since our investment was 5 coins, this is not a bad return.

Now let's look at the situation where we only hold the three-card Royal. The number of pairs of cards we can select from the remaining 47 is 47 x 46 / 2 or 1081. One of these is clearly a Royal Flush. There are no Straight Flushes, no Four of a Kinds, and no Full Houses. The number of Flushes is the number of ways to pick a pair of hearts out of the remaining 9 and that number is 9 x 8 / 2 or 36. But wait! One of those is the Royal Flush, which we have already counted so the number of Flushes is 35. The number of Straights is the number of ways of picking a King out of the four remaining Kings and a Jack out of the four remaining Jacks or 16 ways. Again, one of these is the Royal so the number of Straights is 15. One can obtain three of a kind by picking two Aces, two Queens, or two Tens. In each case there are three ways to do this, so the total number of three of a kind hands is 9. Two pair can be obtained by choosing an Ace and a Queen, an Ace and a Ten, or a Queen and a Ten, nine ways each for a total of 27. The high pair hands are tricky. There are six ways to pick a pair of Kings from the four remaining and the same is true for the Jacks so there are 12 of these hands. But we can also form a high pair by matching the Ace or Queen. Let's look at the Ace; the Queen will produce the same number. There are 3 ways to pick the Ace and once chosen there are 46 cards remaining. We want to choose the second card so that it doesn't produce any of the hands already counted. That means we don't want to pick one of the two remaining Aces (three of a kind), one of the three Queens, or one of the three Tens (two pair). This leaves 46 - 8 or 38 cards we can use. So there are 3 ways to pick the Ace and 38 ways to pick the remaining card so there are 3 x 38 or 114 Ace - Ace hands. The same is true for the Queen so there are 228 high pairs consisting of Aces or Queens. Adding the 12 hands for the King and Jack we finally obtain 240 high pair hands. The expected return table looks like the following:

If we divide 6955 by 1081 we obtain the expected return by holding only the three-card Royal; that number is 6.434. The three-card royal is the better choice.

Now you see why I chose the weakest Royal. If the correct play is to hold the weakest Royal instead of the four Flush then obviously one should hold any three-card Royal instead of any four Flush.

If one combines the above result with my August article it means that one should always play Royals with one card less than the corresponding Flush. This is an easy rule to remember. See you next month.