Probable Outcomes of Doom

Le Monsieur and I had another short, but very productive session this week working on our training app. This week's lesson was focused on starting hands selection in early position (EP).

There are two major reasons why you should play tight in EP: 1) you're the first to act preflop and you'll be out of position (OOP) on all future betting streets, so you need a relatively strong hand to compensate for these positional disadvantages; and 2) there is an increased likelihood that someone else at the table has been dealt a stronger hand than yours.

I'd like to take a few minutes to work through the math on the second item to show what I mean. For example, imagine that you're dealt KTo under-the gun at a full-ring 9-handed table. This is one of those pretty-looking "trap hands" or "reverse implied odds" ("RIO") hands that get so many beginning players into trouble, especially from EP. Here's why it's such a bad hand from up front:

The hands that dominate KTo are: AA-TT, ATs+, KJs+, ATo+, KJo+.  In addition, the pocket pairs 99-22 all have higher equity preflop than KTo, and KTs also has higher equity (52.5% to 47.5%). Here's an image that shows this range of hands that have higher equities than KTo:

There are 1326 possible unique two card combinations in a deck of cards; i.e., there are 1326 possible unique two card hands that each one of your opponent will be dealt. Of the 1326 possibilities, the aforementioned combined range (22+, ATs+, KTs+, ATo+, KTo+) represents 190 of these.* Therefore, 190/1326 = 14.33%, which represents the probability that a single opponent at your table gets dealt one of these hands that are stronger than our KTo.

Now, 14.33% doesn't necessarily sound like a big number, but you have to remember that there are eight opponents at your table, not just one. What are the odds that at least one of these eight have a better hand than our KTo? Unfortunately, it's not just a simple matter of multiplying 14.33% by 8. The math involved with "and/or" probability questions like this can be quite complex... but there are some tricks we can do to simplify things and make the math easier.

For example, we can determine the probability that a player doesn't have a better hand than ours. Said another way, if there is a 14.33% chance someone has a better hand than ours, then it also stands to reason that there is a 100% - 14.33% = 85.67% probability that the player does not have a higher equity hand than ours.

Now, probability theory says that the total probability of two things both happening are their individual probabilities multiplied together. Same with three things, four things, and so on...  If the probabilities of those events are identical, then we can further reduce all this multiplying to just a power equation of P raised to the power of N, where P=Probability of an event and N=number of instances. I.e., Combined Probability = P^N.

So, getting back to our example, if you are first to act at a nine-handed table (i.e., UTG), then there are eight players who also have cards and have yet to act. The probability that a single one of these players does not have a stronger hand than yours is the aforementioned 85.67%. Each player has this same chance of being dealt a weaker hand than yours, therefore we can apply the Combined Probability equation: (85.67%)^8 = 29%. This means that there is a 29% chance that everyone has hands that are weaker than ours. Or, turning this around and subtracting the chance from 100%, we can say that there is a 100% - 29% = 71% probability that someone at our table has a better hand than our KTo.

In other words there's better than a 7 in 10 chance that at least someone has your KTo hand beaten.

And then you'll be OOP if they decide to play that better hand.

So how do you like that pretty-looking KTo now? Not so much, eh?

All-in for now....

-Bug

*To calculate this 190 value by hand, you have to sum up all the possible combinations of cards. For example, there are 6 individual ways to make a pair of Kings: KcKd, KcKh, KcKs, KdKh, KdKs, and KhKs.  Similarly there are 16 ways to make a non-pair two card combination. Adding all these up for the range shown above equals 190 possible combinations. (Note: a far simpler way to calculate these is to use a program like Equilab, Pokerstove, or Flopzilla to do it for you. If you look toward the bottom of the image above, you'll see a line where Equilab provides both the 190 hand figure as well as the 14.33% value.)

Block and Tackle (and Combinatorics)

We all know to assign ranges and not specific hands when hand reading, right? We don't say, "The TAg villain open raised UTG, and then re-raised our LAggy 3bet (with us holding AhKd) on the button. Therefore he must have Aces." Instead, we should be thinking something like, "The TAg villain opened UTG, so his range is TT+, and AQo. Then he saw us 3bet him on the button, but he knows we're LAggy, so he probably thinks we could have a range anywhere from TT+ to AQ+. He then re-raised us, so we can rule out TT and AQ from his range, as he probably wouldn't 4bet with those. So let's put him on JJ+ and AK. So, how does our AKo stack up against this range?"

Big deal, this is standard hand reading in action, right? Yes, but there's an important step that's missing: factoring in so-called "blockers."  A blocker is a card that we know with 100% certainty that the villain does not have in his hand. In other words, these are cards that a) we hold; b) are on board; and/or c) have been inadvertently exposed by another player or the dealer. 

In this example, we hold AhKd. This means that these two specific cards are "blockers" to hands our opponent can have. In other words, we know with complete certitude that they cannot be in the villain's hand.

Uh, okay. That's pretty obvious, Bug. What's your point?

My point is that from a combination point of view, it makes a  difference to the range we put villain on, and therefore how we can/should play our own hand

Without factoring in our two blockers, we said villain's range was JJ+ and AK. What this actually means is that villain has one of the following hand combinations:

• Jacks (6 combinations): JcJd, JcJh, JcJs, JdJh, JdJs, JhJs
• Queens (6 combinations): QcQd, QcQh, QcQs, QdQh, QdQs, QhQs
• Kings (6 Combinations): KcKd, KcKh, KcKs, KdKh, KdKs, KhKs
• Aces (6 Combinations): AcAd, AcAh, AcAs, AdAh, AdAs, AhAs
• Ace-King (16 Combination): AcKc, AcKd, AcKh, AcKs, AdKc, AdKd, AdKh, AdKs, AhKc, AhKd, AhKh, AhKs, AsKc, AsKd, AsKh, AsKs

Of these hand combinations, we're in a coinflip with twelve of them (i.e., the Jacks and the Queens), we're getting crushed by twelve of them (Aces and Kings), and there are 16 we're likely to chop with if we get to showdown (Ace-King)*.

Ah, but we forgot to factor in our Ah and Kd blockers. In other words, we can remove all the Ah and Kd cards from the aforementioned combos. If we do this, we see that our villain actually can only have one of the following hands:

• Jacks (6 combinations): JcJd, JcJh, JcJs, JdJh, JdJs, JhJs
• Queens (6 combinations): QcQd, QcQh, QcQs, QdQh, QdQs, QhQs
• Kings (3 Combinations): KcKd, KcKh, KcKs, KdKh, KdKs, KhKs
• Aces (3 Combinations): AcAd, AcAh, AcAs, AdAh, AdAs, AhAs
• Ace-King (9 Combination): AcKc, AcKd, AcKh, AcKs, AdKc, AdKd, AdKh, AdKs, AhKc, AhKd, AhKh, AhKs, AsKc, AsKd, AsKh, AsKs

Said another way, we've reduced the percentage of the villain's range that is crushing us. It's not a huge amount, but it's enough that it could alter how we play our hand back at villain. Poker is a game of identifying and exploiting small edges, which is exactly what this calculation leads to.

Below are a couple of pie charts that shows this effect in graphical form. On the left is the breakdown of hands without factoring in our two blockers, and on the right is the effect with us factoring in the blockers:

So what do we do with our AK in this situation? Well, that depends of course. How "gambly" is our opponent? What is effective stack size? What game format is this? How much fold equity to we have? And so on. For instance, what if this was a deepstack cash game? If so, we might want to just call and reevaluate on the flop**. If this were early in a moderate-stack tournament, however, we might decide that we're too much of a dog to call and that we don't have enough implied odds, so it's time to muck. If we're getting short-stacked in a fast-structure tourney but still have some fold equity, however, we might just 5-bet shove all-in. It doesn't really matter; the point is that we can make much better decisions if we can make better reads, and one step to doing this is to remove "blocker" cards from the range we put our opponent on.

All-in for now...

-Bug

*I'm ignoring the small advantage villain's suited AKs have over our non-suited AK.

**And remember, the three cards on the flop are also blockers, so we can remove them as necessary from our opponent's hand range, too, at that point as we further refine our read. The turn and the river also are blockers.