Does This Make Sense?

This is a column for “low intermediate” players and it involves 4-card inside straights in games where you get your money back for a pair of jacks or better, same payout for two pair, and 4-for-1 for straights. It also involves a paradox of sorts.

There are a lot of these games — Double Double Bonus, White Hot Aces, Triple Double Bonus, Triple Bonus Poker Plus, Super Aces Bonus, and others. What I’m talking about today does not work for games where you get double money for two pair (i.e. Jacks or Better or Bonus Poker), nor does it work for games where you receive 4-to-1 for straights (i.e. the best versions of Double Bonus.)

That last sentence should have caused many of you to do a double take. I said 4-to-1 rather than the more common 5-for-1. They are equivalent, but often players are a bit loose with their terminology. When a pay schedule lists numbers, the returns are always “for 1.”

Here’s the paradox:  Holding 4-card inside straights with no high cards (e.g. 4578 of mixed suits) is eligible to be held in these games.  (“Eligible” means better than throwing everything away. There may or not be higher paying options in the hand.) Assuming you’re playing dollar 9-6 Double Double Bonus, five coins at a time, holding this inside straight is worth $1.70. Holding 4-card inside straights with one high card (e.g. QT98 of mixed suits), worth $2.02 in the same game, is never held.

Both inside straight draws have four cards to fill them in. When there’s a high card involved, there are also three chances to pair up that high card. Of the 47 possible draws, three extra chances to get $5 (the payout in this game for a high pair), add 3 * $5 / 47 = 32¢, which is the difference between $2.02 and $1.70.

I call it a paradox because the greater of the two hands is never held and the lesser of the two hands is held sometimes (depending on the fifth card). So, what gives?

If you haven’t seen or heard this paradox before, I strongly urge you to see if you can figure it out for yourself before you read on. I’ll wait. No matter how many video poker paradoxes I write about, there are hundreds more that I will never get to. If you’re going to become a decent player, you’re going to have to learn to think for yourself.

The key to the paradox is using absolute values to deflect attention from relative values. The $1.70 and $2.02 numbers are absolute values — that is, what the position is worth on average if you play it out zillions of times.

What is important in video poker, however, are relative numbers. In the hand 4♠ 5♥ 7♣ 8♦ 2♠, we’ve already said the value of holding 4578 is $1.70. The important thing is that the value of throwing everything away is $1.62. Those are the best two choices, and the better of these two is to hold the 4-card inside straight.

From Q♥ T♥ 9♣ 8♦ 3♠, we’ve said the value of holding the 4-card inside straight is $2.02, but the value of holding the queen by itself is $2.12, and the value of holding the suited QT is $2.23. Relatively speaking, the value of $2.02 is less than the value of holding either the single high card or the high card with a suited ten.

Every time there is a 4-card inside straight with one high card, there is necessarily a high card by itself in the same hand — and sometimes a high card with a suited ten. So, we’re never going to hold this inside straight.

The fact that holding this inside straight is better than throwing everything away is interesting, perhaps, but irrelevant. We’re looking for the play with the highest EV, and throwing everything away isn’t going to qualify when there’s a high card in the hand.

Once you realize that we’re comparing one inside straight to throwing everything away, and the other inside straight to something else entirely, the paradox disappears.