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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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Dec. 09, 2008

Holding Kickers

I wrote in a Casino Player article that from AAAK2 you hold AAA in 9/6 Double Double Bonus and AAA2 in 9/6 Triple Double Bonus. I received an email from a reader who couldn't understand why. This is a good spot to teach this lesson.

In both games, we first need to look at how often various things happen if we hold AAA and if we hold AAA2. Let's look at the following:

Drawing Two Cards to AAA


	A	B	C	D	E
		9/6 DDB	9/6 DDB	9/6 TDB	9/6 TDB
			Product		Product

Four Aces w/2,3,4	11	2,000	22000	4,000	44000
Four Aces	35	800	28000	800	28000
Full House	66	45	2970	45	2970
Three of a Kind	969	15	14535	10	9690

Total	1081		67505		84660
Average			62.44681		78.31637

In Column A, we see how frequently we get each of the four possible hands --- namely aces with a kicker, aces without a kicker, full house, or three of a kind. These are the only possibilities when drawing two cards from AAA --- and this will remain the same for both games. When we draw two cards in a 52-card game, we always have 1,081 different combinations we can draw.

To get the numbers shown in Column A, choose the game in Video Poker for Winners, click on ANALYZE - SELECT SPECIFIC CARDS and enter in the hand in question --- perhaps Ah Ac Ad 2h Kd, although the actual suits don't matter in this particular example --- and then click on ANALYZE THIS HAND and then on SHOW DETAILS. Some other software products will produce these numbers. Some won't.

Column B is the relevant part of the pay schedule for 9/6 Double Double Bonus. Column C multiplies the numbers in Column A by those in Column B. (This is called the product). When we sum the products and divide by 1,081, our answer is 62.45. This is the 5-coin expected value (EV) of drawing two cards to AAA.

Column D is the pay schedule for 9/6 Triple Double Bonus. Notice you get more for aces with a kicker and less for three of a kind. Column E is the product of Column A and Column D, and the 5-coin EV is 78.32.

So far we can't tell what to play. To do this, we need to look at the results when we draw one card to AAA2.

Drawing One Card to AAA2

	A	B	C	D	E
		9/6 DDB	9/6 DDB	9/6 TDB	9/6 TDB
			Product		Product

Four Aces w/2,3,4	1	2,000	2000	4,000	4000
Four Aces	0	800	0	800	0
Full House	3	45	135	45	135
Three of a Kind	43	15	645	10	430

Total	47		2780		4565
Average			59.14894		97.12766

Column A represents the frequency of hits. Since this is now a 1-card draw, it will add up to 47. Columns B and D are the same pay schedules found previously, and Columns C and E represent the products comparable to those in the first table.

When we sum the products and divide by 47, we get a 5-coin EV of 59.15 in 9/6 DDB and 97.13 in 9/6 TDB.

To determine the correct play we must compare the return. In 9/6 DDB, 62.45 > 59.15, so we hold the aces by themselves. In 9/6 TDB, 78.32 < 97.13 so we hold AAA2.

Calculating EVs this way is very common. Computer programs like "Video Poker for Winners" and others do this automatically for you. If you didn't know how the numbers were generated in that program, hopefully it's clearer now.

Some players prefer a "less mathematical" explanation of how come the correct play changes. The biggest reason why the play changes is the higher return for Four Aces With a Kicker in TDB. Holding the kicker guarantees that when you do get the four aces, the kicker is automatically included.

The "same" problem could come up with different answers with a different starting hand. I suggest you analyze the following hands using Video Poker for Winners: AAA23 (where you have two kickers), AAA22 (where you have a full house). If you do this exercise you'll see changes to the numbers in the two Column As in this article.

These differences won't affect the correct play this time. When similar differences in other hands do affect the correct play, these differences are called "penalty cards." Once you get the hang of looking at the details in the computer software as shown here, understanding penalty cards becomes much easier.