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VIDEO POKER
 Bob Dancer writes a video poker column for beginners to experts. He also writes a column each week with Jeffrey Compton titled Player's Edge, which features information on promotions at various Las Vegas Hotel. Player's Edge is published each Friday in the Las Vegas Review-Journal. Click here to send Bob Dancer an e-mail.
Feb 12, 2002
Breaking Two Pair in Double Bonus and Double Double Bonus
In 10/7 Double Bonus, you give the average player a hand such as Kh Ks 7h 7s 3d and he will hold the kings and draw three cards. His logic is that a pair of kings only gives you your money back, and two pair only gives you your money back, so this looks like a free shot to get four kings. Easy to understand logic. But wrong.
Let's look at the numbers, for a dollar player. If you hold KK77, you have 47 possible cards you can draw. [The reason it is 47 is that you started with 52 cards in the deck and you are looking at five of them on the initial deal. 52 minus 5 = 47] Four of those cards (the other two kings and the other two sevens) give you a full house. The other 43 cards you can draw give you two pair you started with. If you multiply it out, it is [4 * $50 + 43 * $5] / 47 = $8.83. That means on average, every time you draw a card from this position you will get $8.83. You will usually end up with $5 and occasionally end up with $50, but the AVERAGE is $8.83.
When you draw three cards to KK, it is a bit more complicated because there are 16,215 different combinations of cards you can draw. It was possible to work out the one card draw with paper and pencil, but accurately figuring out the possibilities when there are 16,215 with paper and pencil is too much for everybody this side of Rainman. But with a computer, it is easy. Out of the 16,215 chances, 11,520 of those combinations leave you with a pair of kings worth $5, 2629 of those combinations leave you with two pair which are also worth $5, 1852 times you get three of a kind worth $15, 169 times you'll get a full house worth $50, and 45 times you'll end up with four kings worth $250. If you work this all out, you'll find that the average return from holding the kings by themselves is $7.29.
So holding the kings by themselves instead of two pair is an error worth $1.54. As errors go, this one is QUITE LARGE, but many players make it over and over again several times an hour. No wonder they think the casino always wins!
Instead of kings and sevens, let's now look at aces and sevens. Since we know that drawing one card to KK77 is worth $8.83, how much do you think drawing one card to AA77 would be? Higher, lower, the same, or you don't have a clue?
The answer is the same $8.83. Four times you will get a full house worth $50 and 43 times you will keep the lousy two pair you started with worth $5.
If you draw three cards to AA, however, your results will be different than when you draw three cards to KK. Not the FREQUENCY of the results (you will still get a high pair 11,520 times, two pair 2629 times, etc., down to 45 wonderful times you will get four aces), but the VALUE of four aces is worth $800 and the value of four kings is only worth $250. To find out how much this changes the hand we multiply $550 (i.e. the difference between $800 and $250) by 45 (the number of occurrences) and then divide by 16,215 (which is the total number of combinations possible when you draw three cards.) This comes out to be $1.53. Another way of saying this is that holding AA77 is about a penny better than holding just AA.
The value of the full house is part of this equation. So if full houses only played $45 instead of $50, AA would be a better play than AA77 by 36˘, but KK77 would still be better than KK by $1.17.
For Double Double Bonus, the conclusion is different. In this game, you ALWAYS prefer AA to AA77, and ALWAYS prefer KK77 to KK. The value of AA will change depending on the other three cards. For example, in the game where full houses pay 50, holding AA77 is still worth $8.83, but the value of AA is $9.70 when the other three cards are 887, $9.63 when the other three cards are 883, $9.56 when the other three cards are 445 and $9.48 when the other three cards are 443. The reason for differences, of course, is that four aces with a 2,3 or 4 kicker is worth $2,000 and four aces with a 5-K kicker is worth "only" $800. So the more good kickers you "burn" BEFORE the draw, the fewer chances you have to end up with the good hand AFTER the draw. But all of these values are significantly above the $8.83 you would get if you held AA77 (or AA33 for that matter, since kickers don't matter when we are talking about full houses).
If the amount of the full house only pays $45 or (horrors!) $40, the numbers change, but the strategy rule doesnąt. For Double Double Bonus, break two pair including aces, but hold two pair including kings, queens or jacks.
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