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.
Oct. 08, 2002
Which Two-Pair Hand is More Frequent?
Two weeks ago I said that there are 123,552 different two pair hands, including suits for all five cards. I received an email from a reader who told me that since two pair was a FINAL hand every 7.735 hands in 9/6 Jacks or Better, was it fair to multiply these two numbers and conclude that each specific two pair hand comes around every 955,674 times? If so, he claimed, it would be significantly easier to be DEALT a royal flush than to end up with a specific two pair hand.
I'm sure his conclusion is correct, as a dealt royal occurs slightly more often than once every 650,000 hands. However it must be that certain two pair hands are more frequent than others. I hadn't thought about this before, but found a number of patterns. Every game has a slightly different pattern of two pair hands, so we are limiting this to 9/6 Jacks or Better and we assume you are playing according to computer-perfect strategy. Take this as a quiz and see how you do.
Which is more common? And there are no ties. With every choice, there is an easily understood reason why one final hand occurs more frequently than the other.
a. KK553 or 99553?
b. KKQQ3 or KK883?
c. 99887 or 99447?
d. TT66A or TT66J?
e. JJTT5 or KKTT5?
f. AA553 or KK553?
g. 55443 or 44332?
Below are my answers. It's possible I've missed an important relationship that will change one or more of these answers. If so, please let me know at bobdancer@lvcm.com.
a. KK553 is more common than 99553 by a big margin simply because it is correct to start a hand with KK far more often than you start a hand with 99. Letting parentheses indicate same suit, from (AQJ)KK you should hold the kings but from (AQJ)99 you should hold the 3-card royal. And from (K973)K you should hold the high pair but from (K973)9 you should hold the 4-card flush. In addition, often you hold a single K and draw four cards --- but you never hold a single 9 and draw four.
b. KKQQ3 is more common than KK883 for a couple of reasons. First of all we have exactly the same logic as in the previous paragraph. The second reason is that there are many hands where it is proper to draw three cards to either (KQ) or KQ, but it is NEVER correct to draw three cards to K8.
c. 99887 is more common than 99447 because there are 3-card straight flush draws including a 9, 8, and 7. For example, from KQ(987) it is correct to hold (987). But from the equally likely KQ(974) it is correct to hold KQ.
d. TT66J is more common than TT66A. Why? Because there are many hands where holding (JT) is correct [such as (JT) 872] but there are NO hands in this game where holding AT is the correct play.
e. JJTT5 is more common than KKTT5. It is correct to draw three cards to (JT) more frequently than it is to draw three cards to (KT). Compare the equally likely (JT)A37 with (KT)A37. In the first hand you should hold (JT) but in the second AK. Or maybe compare (JT5)93 with the equally likely (KT5)93. Good players play (JT) in the first hand and K in the second. Finally, compare the equally likely (JT)K53 with (KT)J53 and you'll hold (JT) in the first hand and JK in the second.
f. AA553 is more common than KK553 because you sometimes draw two cards to (A53) but you should never draw two cards to (K53).
g. 55443 is more common than 44332. This might be the toughest one to work out. The 3-card straight flush (543) is more powerful than the 3-card straight flush (432) so you hold it more. Compare, for example, [KQ](543) where you hold (543) with [KQ](432) where you should hold [KQ]. For the logic behind this, see my June 18, 2002 column.
How'd you do? I suspect that only experts would get these all correct as you needed to know correct strategy pretty well to get some of the answers. Is the information in this column immediately useful? Absolutely not! I hadn't even considered these things until recently and it hasn't slowed me down too much. I believe, however, that the more you know about these games the better you will succeed at them. One of the secrets to my personal success is that puzzles like this fascinate me and so I'm better able to figure out new puzzles (such as how to take advantage of a particular promotion) easier than many others can. I suggest the more you work at these puzzles, the better your gambling results will be too.
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