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December 2 — Understanding the concept of conditional probability is the key to late street hand reading and weighted combinatorics

Conditional probability is a great way to describe the process of hand reading. When we hand read, we are trying to give our opponent a range of hands based on an assertion or evidence (the action). For example, if we raise six times the big blind over a couple of limpers from the button and a tight player calls from the small blind, we can discount a very large portion of hands from his range, which are much different than if the pot was limped around. “If he calls $65 in the small blind, then—”. So, if the flop were to come out 7-5-3 rainbow, we could easily discount the fact that this player flopped a straight because of this “if, then” (raise and only call with a tight range) assertion. We can take this one step further, as we have more evidence when we plug in the action post-flop. If the small blind led out on the flop and was called by two people and then checked a 6 on the turn, it is highly unlikely that he ever has a straight, because he would not play 6-4 or 8-6 preflop for a raise and he would not lead on the flop with 4-4 and check the turn when 4-4 made a straight.  

This conditional probability hand-reading technique is not at all hard to master, but people freeze in the moment all of the time and do not consider the previous street action when doing an accurate range analysis.  

Let’s take a look at a hand that a subscriber of my site, CrushLivePoker.com, phoned in on my weekly call-in podcast, Crush Live Call-Ins, free for anyone to listen to at 7:45 p.m. ET every Sunday. This spot occured in a $3-$5 no-limit game with effective stacks of about $900. My student was on the button and cold-called a $20 raise from the under the gun (UTG) player with A 3. The big blind also called and they saw a flop of A 7 3 three ways. The big blind checked and the UTG player made a continuation bet of $35. My student rightfully recognized that he needed to build a pot up with stacks this deep with two pair and raised it to $100 (actually smaller than I would have made it). The blind folded and the UTG player called. The turn brought out the K, completing the rainbow, and now the UTG player suddenly led out for $60. We both found this strange because usually if the preflop raiser was going to play a stop and go on a safe turn card it would be on a flop that contained some sort of draw. Of course, with A-7-3 rainbow, the board is about as dry as it can get. I recommended to the caller to just flat with the A-3 as to not overplay his hand and get all weaker single-paired aces to fold and only better than his aces up to call. He indeed did flat and they saw the river bring the A. At this point, the UTG villain now led out for $220 and the question became whether or not we should raise with A-3 on the river. My student thought that it should be a raise-fold situation because of the value he could get from pocket kings or pocket sevens.  Although I agree that both of those hands would definitely call a raise at the end if we use conditional probability and weighted combinatorics to aid us in hand-reading, we can see that both of these hands are very unlikely. 

Let’s first take a look at the possibility of the villain in the hand arriving with K-K on the river. Here is the evidence followed by my assertions:

A. Preflop the UTG raised to $20
B. On the flop, UTG bet $35 in to two opponents on an A-7-3 rainbow board
C. UTG called a $100 raise from the button after continuation betting. 
D. UTG makes a weak lead on the turn after hitting a very hidden set. 

So, what are the chances of A, B, C and D happening all together? We can actually evaluate this by looking at the probability of each event individually through what if statements then multiplying them all together to get the total probability of the whole event.  

So for the sake of simplicity let us say that assertion A happens 100% of the time (UTG opening K-K to $20 preflop) and we can represent that by the number 1.
 
For B what are the chances that the UTG continuation bets an A-7-3 rainbow board with pocket kings into two people? This is just an estimate, but I do not think that that this will happen all that often. I give it about 30%, represented by .3.

Now we look at C. What are the chances that after continuation betting a dry board with kings, he would call a $100 raise? This really is the most telling part of the hand. I’d optimistically give this 30%, represented by .3.

Lastly, let us examine D. What are the odds that a player would lead out for a one-quarter pot sized bet after hitting a sneaky set on the turn? If the opponent called the flop raise he must have thought that the button was bluffing at least some of the time or would fire again with a value hand. I think he would check raise here far more often than he would weak lead. For simplicity sake again, I will give the chance of this happening another 30%, represented by .3.  

So we have an equation of A at 1, B at .3, C at .3, and D at .3. If we multiply these numbers together we get .027 or 2.7% he gets to the river with kings. If we look at the total number of combinations of pocket kings that there are on the turn, three, we multiply them by 2.7% and get less than 1/10 of one combination.  

Now, let us do this exercise with A-K. Let us express A as .9, B as .9, C as 1, and D as .3. So, for this situation of A then B then C then D, we get about .24 for 24% of the combinations. Since there are also three combinations of A-K, we can see that the villain showing up with that hand is about nine times more likely or about .72 total combinations. 

Now let us run through 7-7. Let us say A is .5, B is .4, C is .8, and D is .2. If you multiply all of these probabilities out, we get .032. Again, there are three total combinations of 7-7, so we end up with just under one tenth of one combination. So, for hands that we beat, K-K and 7-7, we have about .2 combinations, and for hands that we lose to about .72 combinations. If we need to be good over 50% of the time when called, you can see here that raising clearly is not the right play.

On the surface, this process may seem complex but if you can practice the theory behind these concepts, you will become a much better hand reader, which will lead you to superior decisions.  ♠

Follow Bart for daily strategy tips on Twitter @CrushLivePoker and @BartHanson. Check out his poker training site exclusively made for live cash game play at CrushLivePoker.com where he produces weekly podcasts and live training videos.