Conditional Probability

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

From Wikipedia--

"In probability theory, a conditional probability measures the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred."

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 to 6x the big blind over a couple of limpers on the button and a tight player cold calls from the small blind we can discount a very large portion of hands from his range, different than if the pot was limped around. "If cold call $65 in the small blind, then--". So if the flop were to come out 357r 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 turn it is highly unlikely that he ever has a straight because he would not play 46 or 86 preflop for a raise and he would not lead on the flop with 44 AND check the turn when 44 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 us 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 745PM 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 UTG with A♦ 3♦. The big blind also called and three ways they saw a flop of A♠ 7♥ 3♣. The BB checked and the UTG 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 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 A73r the board is about as dry as it can get. I recommended to the caller to just flat with the A3 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 a river of the A♣. At this point the UTG villain now led out for $220 and the question became whether or not we should raise with A3 on the river. My student thought that it should be a raise-fold situation because of the value he could get from pocket Ks or pocket 7s. 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 us first take a look at the possibility of the villain in the hand arriving with KK 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 A73r board

C. UTG called a $100 raise from the button after cbetting.

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% (UTG opening KK to $20 preflop) and we can represent that by the number 1.

For B what are the chances that the UTG cbets an A73r board with pocket KKs into 2 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 cbetting a dry board with KK he would call a $100 raise? This really is the most telling part of the hand. I'd optimistically give this 30%.

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%.

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 KK. If we look at the total number of combinations of KK that there are on the turn, 3, we multiply them by 2.7% and get less than 1/10 of one combination.

Now, let us do this exercise with AK. 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 3 combinations of AK we can see that the villain showing up with that hand is about 9 times more likely or about .72 total combinations.

Now let us run through 77. 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 77 so we end up with just under one tenth of one combination. So for hands that we beat, KK and 77, 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.