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Bob Dancer writes a video poker column for beginners to experts. He also writes a column with Jeffrey Compton, "Player's Edge", featuring information on promotions at various Las Vegas casinos. Player's Edge is published each Friday in the Neon section of the Las Vegas Review-Journal. Click here to send Bob Dancer an e-mail.

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Jan. 23, 2008

A Simple Penalty Card Lesson from 9/5 Triple Bonus Poker Plus

As discussed in a recent column Triple Bonus Poker Plus (TBPP) is a variation on Double Bonus Poker with the following pay schedule for the best version:

Royal Flush 800
Straight Flush 100
Four Aces 240
Four 2s thru 4s 120
Four 5s thru Ks 50
Full House 9
Flush 5
Straight 4
Three of a Kind 3
Two Pair 1
Jacks or Better 1
Let's look at the following three hands:
a. 4h 5c 6c 7c 9s
b. 4h 5c 6c 7c Ts
c. 4h 5c 6c 7c Js

In all three cases, we have a perfect 3-card straight flush (i.e. '567'), a 4-card consecutive straight with no high cards (i.e. 4567), and a "throwaway" card (i.e. 9, T, or J).

If I told you that in all three hands you should hold the 4-card straight, your response might be, "Okay. So what?" You probably know that it is typical to hold the 4-card straight in most games, but this game returns more for the straight flush so you might not be sure unless this is one of your main games.

But what if I told you that in only one of these hands it was right to go for the straight flush? My question to you would be, "Which hand and why are they different?"

It's easiest to see that the 3-card straight flush is weakest in hand (a). The nine is a straight penalty to the '567', meaning it is close enough to be in the same 5-card straight. For the 5-coin $1 player, this particular straight penalty reduces the value of '567' by about 7.4¢.

So our "correct" answer has to be (b) or (c). Neither the ten nor the jack is a straight penalty. Additionally they are not flush penalties. Since these are the most common two types of penalties, these are all some people know.

There is something, however, that makes the value of '567' 1.4¢ less in example (c) than it is in example (b). Can you figure out what it is? It's not difficult, but it's a type of penalty card that isn't discussed very frequently.

The answer is that the jack in example (c) is a "high pair penalty." Drawing two cards to '567' in example (b), you're going to end up with a high pair 24 times out of the 1081 possibilities. That is, you can draw six different pairs of jacks (JsJc, JsJd, JsJh, JhJc, JhJd, and JdJc), or six different pairs of queens, kings, or aces.

In example (c), you only end up with a high pair 21 times out of 1081 because the first three pairs of jacks listed are not possible because you threw away the Js before the draw.

Both of my statements at the beginning were correct. That is, it is correct to go for the 4-card straight in all three cases AND in one of the hands it is correct to go for the straight flush. Although these two statements seem to contradict each other, in example (b), whether you hold 4567 or '567', the Expected Value is exactly the same. Holding 4567 is less risky (i.e. it has a lower variance), however it is also less exciting. The highest valued result you can get from holding it is $20. Holding '567' could end up with a $500 payday, although you end up with a score of $0 about 87% of the time from this starting position.

I've heard it argued that even though the EVs for the two plays are identical, holding 4567 is "better" because it has a lower variance. Is this just the personal preference of the person making the argument, or is there enough mathematical justification to this position that it makes sense for the rest of us to adopt that policy as well?

The truth of the matter that if you ALWAYS look to reduce the variance of your plays, you will reduce your chances of going bankrupt. Playing conservatively (where I'm using the word in a financial sense, not a political sense) is sensible. Avoiding "gamblers ruin" is a legitimate goal, although the effect is infinitesimal in this particular situation.

A better reason to go for the straight on this hand is that when you do, the fifth card is truly a "throwaway" card. If you ALWAYS extend 3-card straight flushes into 4-card consecutive straights when you can (EXCEPT for 'JT9' 8), the game becomes simpler. If you need to closely examine whether the fifth card is less than a jack (so the high card penalty doesn't take effect) and is far enough away from the 3-card straight flush that it can't be part of the same straight, it will slow you down a little bit.

Assuming you're playing the game with a slot club benefit of at least 0.20%, the more hands you can play, the more money you can make. Therefore, playing the simpler, faster strategy is superior.