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Winning Concepts in Draw and Lowball

by Mason Malmuth

The first question is what hands will your opponent make when he draws two? This is answered by Table I below.

Before we go on, I want to point out that this table is accurate no matter what the two card draw may be. For example, if you were to draw to 7,6,5,Q,K (throwing away the queen and king) the chances of making a 7 is the sum of the 5, 6, and 7 entries in the table or 0.0922.

All other probabilities are the same.

So, what is the minimum hand that we can bet for value? First, lets assume that our opponent will not raise bluff after we bet. That is, if he raises, we throw our hand away. Second, let's assume that our opponent does not have a tell that indicates whether we should check or bet. In other words, our decision is based on mathematics only.

Since we want our bet to be profitable, lets agree to only bet if we feel that our win rate (on this bet), when called, is at least 50 percent. This means we bet with the top 50 percent of our opponents calling hands.

Table II below is a repeat of Table I. Except that this time the totals are cumulative. That is, the totals are for the finishing hand or better. Also given are the 50 percent points which will correspond to our minimum bet hand.

Now, for example, suppose you knew your opponent did not have the joker and you believe that his minimum calling hand is a queen. What hand can you bet?

Looking at the 50 percent point in the table above we see .2532 which is approximately equivalent to him making a 10-5. (These calculations are not presented.) Hence we can bet any 10-5 or better for value.

Table III below gives minimum value betting hands for your opponent's corresponding minimum calling hand. Also note that since your opponent did make a "loose" play by drawing two cards -- this does not mean that his play was incorrect -- he will often drop his calling requirements to counter your bluffs.

Two final notes. Just because your opponent has a high calling requirement does not mean you can't bet. You just cannot value bet very often, but many bluffs will be successful. Also, if you think your opponent may raise as a bluff more often than he will raise legitimately, value bet more hands if you intend on calling the raise. Similarly, if you think your opponent may raise as a bluff less often than he will raise legitimately, value bet less hands if you intend on calling.