Pocket Hand Win Frequency

            The inspiration for this week's column comes from a reader this week who e-mailed me and had a question regarding a column I recently wrote about Ultimate Texas Hold'em.   In that column I was describing a hand I saw actually played at the Wynn a while back.  The Player was dealt a K-8 offsuit and did NOT wager 4x at that point, instead opting to wait for the Flop.  I had said that this hand was a marginal 4x wager and that the Player could expect to win the hand 54%, while losing it 42+% of the time. 


            My reader wanted to know how I came up with these numbers.  The methodology is rather simple.  I dealt each of the possible 2-card combinations to one Player and then dealt 7 random cards (2 to the other Player and 5 community cards).  I did this 100,000 times for each possible pocket hand.  100,000 hands might be a little light for top precision, but I did this to analyze UTH and only need a good ballpark to get me close to the right strategy.  It doesn't matter if AA wins 84.96% or 85.01% of the time.  It is clearly a 4x Wager in UTH.


            With a hand like K-8 offsuit, a little more work was necessary.  Some of you who have some knowledge in gaming math may wonder about this.  54% is a long ways from 50% which is frequently our trigger point for making a wager.  But, not in the case of UTH.  Ultimate's unique betting structure where the Player may make a larger wager earlier in the game, but then cannot wager again changes our math a little bit.  In fact, there is no absolute number we are looking for.  We just know it is ABOVE 50% and can't possibly be below it.


            In UTH, you have a choice of wagering 4x after only seeing your pocket cards or checking at that point and seeing the community Flop first.  At that point, you may wager 2x or check again, waiting to see the Turn and the River.  When you are dealt a Pair of Aces, you definitely want to wager 4x.  Yeah, the Flop might be really ugly for you, like a suited 567 in a suit that neither of your Aces are.  But the odds of this are rare and you are better off wagering 4x and risking the times the Flop is bad over waiting to see the Flop and wagering only 2x.  As I stated earlier, the Player will win with AA around 85% of the time.  You're better of taking that and running with it rather than seeing the Flop and limiting yourself to wagering 2x.  Sure, you might wager 2x and win 90% of the time and hold off and wager only 1x for the other times, but in the end, you won't win as much money.


            With the K-8 offsuit, it is not as clear.  You're only going to win 54% of the time.  Are you better off waiting to see the Flop and then deciding to wager 2x or to check again.  If the Flop contains a K your hand gets much stronger (especially if there is no Ace on the board).  You're much more likely to win the hand at this point, but you'll be limited to wagering only 2x.  At the same time, the Flop might be A-Q-9 and be of very little help to you, and put you in a position where not even a K or an 8 on the Turn/River will win it for you.  Are you better off waiting for this extra information before you make the decision?


            Normally, when the Player is more likely to win than to lose it pays to make the additional wager.  But that is when it is an optional wager that doesn't impact further chances to wager.  In UTH, it does impact the ability to wager later and you get to wager more NOW than later.  So, we know that if the probability of winning is lower than the probability of losing there is chance that you will want to wager 4x.  But, if you are only slightly more likely to win that to lose, what do you do then?

            The answer is, you let the expert do all the heavy lifting for you.  After I ran my 100,000 hand simulation for each pocket hand, I sorted them by the win frequency.  Any hand that the Player would lose more often than win was clearly not 4x and I eliminated them.  I then looked at the hands that were about 5% more likely to win than to lose and I created a program that played UTH randomly and set up the strategy at that point.  I played 10 million hands.  Then I tried at 4% and 6% and then at 3% and 7%.  There are only a handful of hands that were changing strategy each time.  After a few simulations, I found my strategy.  To be sure, I refined it a bit further, trying specific hands that were close to the line.  The results were fairly clear.


            It was from this that I determined that K-8 offsuit is worthy of the 4x.  But, it should be noted that I say 'marginal'.  We know from watching people play UTH that many are a bit nervous about wagering 4x.  It is a lot of money to put on the line.  When you're dealt AA or KK or AK, you feel a bit more bold about it.  With K-8 offsuit, many Players can't bring themselves to do it.  Are you killing yourself by chickening out?  Not really.  You are definitely costing yourself some money in the long run.  If K-8 offsuit was the only case of this, you'd likely barely notice.  This hand will happen just under 1% of the time.  Once in about 110 hands or once in 3-4 hours of play.  The net cost of playing it 'safe' might be just a few pennies in payback. 


            The problem comes in if you start doing this for a lot of the marginal hands and how you define marginal.  A suited K-8 will win 57% of the time (and lose 40%) of the time and really isn't so marginal.  A Player should be wagering 4x nearly half the time.  If you do it 40% of the time (the 'right' 40%), you'll cost yourself a little, but you'll need a smaller bankroll to get by.  You drop it down to 25% and your bankroll can start a little smaller, but is likely to end a LOT smaller!