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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 26, 2006

Another Look at Penalty Card Error Report

A few weeks ago I discussed how Video Poker for Winners lists all of the exceptions to the penalty-free strategy. In that article, available a few clicks away, I discussed the two cases in 9/6 Jacks or Better when it's right to hold a 4-card flush over a 3-card royal.

When I was studying the game, I learned that when you have a suited 'QT' in the same hand as an ace of a different suit, you hold 'QT' when there is no flush penalty (i.e. another card suited with the 'QT'), and hold AQ when there IS a flush penalty.

Since this is a simple rule, I was surprised the first time I saw that this one rule took up three separate lines in the error report, specifically lines 9, 10, and 11.

	Example	Strategy	Perfect	Strategy	Number	Total
	Hand	Play	Play	Error	of Hands	Error

1	TH TD 2H JH AH	RF3 'AHT'	FL4 2h	0.00462535	288	1.33210080
2	2H TH JD QH AH	RF3 'AHT'	FL4 2h	0.03700280	576	21.31361280
3	9H JD QD KH AH	RF2 'QJ'	ST4 4h1i	0.01140920	108	1.23219360
4	2H JH QH KD AD	RF2 'QJ'	ST4 4h1i	0.06012950	216	12.98797200
5	8H 9H JD QH KC	SF3 1h2i	ST4 3h1i	0.04625345	96	4.44033120
6	2D 3H TH JH KD	RF2 'JT'	HC2 KJ	0.01017575	720	7.32654000
7	2H 7D TH JH KD	RF2 'JT'	HC2 KJ	0.02991060	180	5.38390800
8	2H 8D TH JH KD	RF2 'JT'	HC2 KJ	0.04964540	360	17.87234400
9	2D 3H TH QH AD	RF2 'QT'	HC2 AQ	0.02251000	1080	24.31080000
10	2H 8D TH QH AD	RF2 'QT'	HC2 AQ	0.04224485	216	9.12488760
11	2H 9D TH QH AD	RF2 'QT'	HC2 AQ	0.06197965	216	13.38760440
12	2D 3H 9D TH KH	RF2 'KT'	HC1 K	0.00773695	1440	11.14120800

	Error in Coins					129.85350240
	Error in Strategy					0.00099927%

All three of these examples include the flush penalty, but the second and third also include a straight penalty. The second example (line 10) includes an 8 penalty and the third example (line 11) includes a 9 penalty.

As a player, I don't particularly care about this distinction. The flush penalty is all I look for. If the straight penalty changes the strategy, I'll look for that too. But since it doesn't in this case, I ignore it.

So even though humans don't need to consider whether or not there's a straight penalty, the computer does. Why? Because each of the various types penalty card affects the value of 'QT to a different degree. While players don't need this information (this time, at least) in order to play perfectly, the information is necessary to determine the total error in the strategy.

Each time there is a flush penalty without a straight penalty, AQ is worth 2.25¢ more than 'QT' for the 5-coin dollar player. When there is an 8 penalty, there's a 4.2¢ error, and a 9 penalty is worth 6.2¢.

Counting the number of times each of these occurs isnâ€™t particularly difficult once you learn the technique. For line 9, there are 4 suits for the 'QT', three suits for the ace once the suit of the 'QT' is decided, 6 possible cards for the flush penalty (2, 3, 4, 5, 6, or 7), five values for the fifth card once the flush penalty is determined, and three suits for the fifth card, as they can be any suit other than the suit of the 'QT'. Multiplying these together, 4 * 3 * 6 * 5 * 3 = 1,080.

For lines 10 and 11, the number of events is one-fifth as large, because instead of any of five possible ranks for the fifth card, there is only one --- namely an 8 in line 10 and a 9 for line 11. One-fifth of 1,080 is 216, or equivalently, 4 * 3 * 6 * 1 * 3 = 216.

The total error in coins at the bottom of the chart (129.85) is the sum of each of individual errors. The way we determine the percentage error is to divide the error in coins by the number of possible hands (2,598,960) times five coins. That is, 129.85 / (2,598,960 * 5) = 0.000999%.

This report contains more information than most players want on a day-to-day basis. But sometimes some of you are interested in exploring more about the games. (The more you know about the games, the better your results will be.) For those times, it's nice to know the report is available.

Video Poker for Winners! is now available for a free 10-day trial at http://www.videopokerforwinners.com/Default.asp