Saturday, August 12, 2006

Holdem Straight Flush Odds and Strategy

One reason why I haven't posted much lately is that I've been playing a whole bunch of limit poker. I have discovered a repertoire of plays, based totally on probability, which allow me to win playing online limit poker. The basic notion is that one must be very disciplined, play very tight, and not make any pre-flop mistakes to win over the long run.

One thing I've wondered about, probably like many other students of the game, is whether one should pre-flop raise in anticipation of making a straight or flush draw hand? If this is a worthwhile play, one that most pros already know, I should add it to my toolbox of techniques, and make even more money. I scoured the web searching for the answer to this question: what are the odds that, given starting hands of suited connectors (SF), one gap connectors (GF), two suited cards (F) or unsuited connectors (S), of flopping a premium (flush or outside straight) draw. These sites came close and told me what I already knew, the odds of completing the draw by the river, or flopping a flush draw:

But I could not find the answer to my complete flop question, so I calculated it (detailed analysis below). The first three columns of this table shows what the odds are of not flopping, flopping and making a draw hand by the river. The last three columns show many other people must be in the hand in order for it to be profitable. As you can see from the table, unless you have premium connectors (JTo or better), it's never probabilistically worth chasing a straight draw, even if you limp in. The number of people is determined by multiplying the cost of the pre-flop play by probability that you don't flop it, and then divide by the probability that you do make it, to give the probabilistic worth of the play. Then calculate the ratio of the total cost of the play to the amount that you need to win to figure out how many players must be in the hand. The effect of pre-flop raising is that it increases the cost of the hand and the number of people that need to be in the hand in order to break even, but it can greatly increase the probabilistic worth of continuing.




































For example, in a $1/2 game, if I decide to raise from late position with a straight flush draw, the probabilistic worth of raising is $2*81.5%/18.4%/51.2% = $17.3. Since it will cost me $5 total to see the river ($2 preflop + $1 flop + $2 turn), assuming no other raises in the hand, there must be at least 2 other people (17.3/5 - 1) in the hand to build a pot big enough for the pre-flop raise play to be worthwhile. So, if two others limp in ahead, this might be a good, profitable play, assuming the limpers call and I don't end up getting outflushed or on the ignorant end of the straight. If I just limp in, I only need to play heads up ($8.65/4 -1) for this play to work out. You may wish to consider calling a pre-flop raiser from early position as well. And if you have large suited connectors (JT or better), which would give you more outs and a greater chance to draw a winning hand, you might even consider re-raising from late position.

The flop answer is determined by dividing the total number of flops by the number of hands that result in a draw, in any combination. Not considering your opponents cards, there are 50*49*48=117600 different flop hands. In a flush draw, for example, there are three possible combinations: either the first two, the second two or the first and third cards make the draw. Since there are 11 cards left in the suit, the number of draw hands are: 11*10*48+(50-11)*11*10+11*(49-10)*10=11*10*[48+39+39]=13860. Calculating the ratio yields the probability that the flop is a flush draw (including the few rare occurrences of flopping an outright flush): 13860/117600=11.7%. For connectors, there are eight cards on either side, and then eight more on either side, to result in an outside straight draw: 8*8*[48+(50-8)+(49-8)]=8384 flops. For gaps, one card must fill the gap, and then eight can wrap around. For a straight flush, I added the number of straight hands, and subtracted the number of straight flush hands from the total: [13860+8380-572]/117600=21668/117600=18.4%.