Why is There a Difference?

You’re playing dollar Double Double Bonus with progressives on both the royal and aces with a kicker (AWAK). On the hands we’re going to talk about today, the only thing that matters are quads and full houses, so assume AWAK pays $2,200 and a full house pays $45. If you’re not familiar with the game, trips pay $15 and aces with the fifth card not being a 2, 3, or 4 pay $800.

I’ll start with the three hands in question, where suits don’t matter:

1. AAA45
2. AAA44
3. AAA43

Here’s the question: How do you play these similar hands? Know before you begin that in one hand it’s right to hold four cards and in the other two it’s right to hold three. Which is which? And, far more difficult, why? (Captain Obvious might say, “It has something to do with kickers.” He’s partially correct, of course, but I would judge that response as not being complete.)

If you make a wild-ass guess, you’ll pick the hand where you hold four cards one time in three. This is a case where, I suspect, if my readers actually try to figure it out, many will do worse than one time in three! Before you start accusing me of being condescending to my readers, know that I would have gotten this question incorrect had I just blurted out an answer before thinking it through.

In case you are unfamiliar with the basic strategy in this game, if the AWAK paid $2,000, the correct play on all hands would be AAA. If the AWAK paid $2,400, the correct play on all hands would be AAA4 (or equivalently on the last hand, AAA3.)

Were I to be given such a question, I would try to see how each of the hands differed from the others. They all include AAA4, where the 4 is a “kicker.” The first one includes a non-kicker, the second one includes another kicker that is paired with the 4. And the third includes another kicker that is unpaired with the 4. So which of those is it?

Got your answer? If not, I suggest you go try to figure it out before you read on. I promise I don’t mind waiting for you.

This is obviously a choice between AAA and AAA4. Holding AAA4, you’re only possible pays are $15, $45, and $2,200. Holding AAA, you can have these outcomes, plus $800.

 Just looking at the value of AAA, it must be highest when you were originally dealt AAA45 because this is the only hand of the three where there are still 11 potential kickers among the 47 undealt cards. (The other two only have 10 potential kickers.) So, since I know that on two of the hands, the correct play is AAA, I’m going to choose AAA45 as one of them.

We still have to choose between AAA44 and AAA43 as to which one you hold AAA4 and which one you hold AAA. Holding AAA4, they both have the same chance of getting AWAK — namely one chance in 47. But they differ in how frequently you can get a full house.

Holding AAA4 from AAA44, there are only two additional fours you can get to complete the full house. Holding AAA4 from AAA43, there are three additional fours you can get to complete the full house. This one extra full house is worth $30 (i.e., $45 minus $15) and it happens over 47 draws. This difference of $30/$47 equals almost 64¢! No wonder the hands are played differently!

Here is a table showing the values of each play in each hand.

	AAA45	AAA44	AAA43
AAA	$64.4820	$63.2146	$63.1869
AAA4	$63.4023	$62.7660	$63.4043
Difference between the plays	$1.0797	$0.4486	-$0.2174

Some of my articles discuss hands where the difference between two plays is 3¢. Here the smallest difference is almost 22¢ and the largest is almost $1.08. These are not trivial amounts, yet before this article most of us (including me) had no idea the differences were this significant.

If you were playing a version of this game where full houses paid 40 instead of 45 (which would only make sense if the royal were really, really high), the plays on all three hands remain the same. All values go down, of course, because the value of the full house is a key part of the equation, but it’s still correct to hold the kicker from AAA43 and not from either AAA44 or AAA45.