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.)

Hand Chart Construction, Part 6: Non-Ace Broadways

In this installment I tackle the remaining non-Ace Broadway cards, also known as "Trouble Hands":

King-Queen. Also known as: Romeo and Juliet, Marriage, King of Queens, and Lucy and Ricky.

• Hand Type: Semi-bluffing connector.
• Position: I will play KQs from EP, but I KQo only from MP+. 
• Stack Sizes: When stacks are shallow, look for flopping TPGK. When stacks are deeper, do not assume your top pair is good if you get action; instead, treat KQ as a drawing hand, looking for two pair plus.
• Table Dynamics: At a LAg table, KQ can often pay off if you play passively. At a tight table, you can and should treat this hand as a semi-bluff hand, but be more than willing to toss it if a TAg gives you action.
• Upstream Action: Unless the effective stacks are very deep and/or the villain is very LAggy, KQo can rarely take a raise, even IP. KQs can frequently take a raise, but you need to proceed cautiously.
• How to Play:  KQ is a dangerous hand that many beginners get married to. It's stronger than a traditional "trap hand" but it's often crushed if you're getting resistance. I suggest you play it strong preflop, but be very cautious postflop.

King-Jack. Kojack, King-John, Tucson Monster.

• Hand Type: Semi-bluffing "trouble" one-gap connector.
• Position: I play KJs from early-MP, but won't touch KJo until late-MP.
• Stack Sizes: Only consider getting it in with KJ if your opponent is very short-stacked, and even then think twice. Deep stack play with KJ is extremely dangerous unless you flop a monster.
• Table Dynamics: TAg and/or LAg tables with this hand are equally difficult. Remember, it's a trouble hand.
• Upstream Action: ABC poker says to just fold this hand to any upstream aggro action.
• How to Play:  KJ is the epitome of the so-called "trouble" or "trap" hand. It looks much, much stronger than it actually is, and if you get to showdown on an action board with this hand, you're rarely going to be good with just top pair. Open-raise in LP, don't call raises or RR's unless there's a damn good reason to do so.

King-Ten. Katie, Big Al.

• Hand Type: Weak semi-bluffing "trouble" double-gap connector.
• Position: I only play this hand in LP unsuited, but I will play in in MP when suited.
• Stack Sizes: Like KJ, this hand is very vulnerable when stacks are deep. And if a pro-shorty shoves on you, just fold. Only get it in PF against the worst players who are extremely short-stacked.
• Table Dynamics: Like KJ, this is trouble regardless of table dynamics.
• Upstream Action: Fold to any upstream raise.
• How to Play:  This hand actually gets people into less trouble than KJ, primarily because it doesn't look as strong; i.e., most players are able to realize this hand is trouble from the start and get away from it. When you get this hand, just ask yourself a) what kind of flop do you want to see; and b) how likely is that flop. More often than not, you're going to just fold it.

Queen-Jack. Maverick, Pinochle.

• Hand Type: Semi-bluffing connector.
• Position: I play QJs from MP, and QJo in LP in unraised pots.
• Stack Sizes: I actually like deeper stacks with this hand than shallow ones, as it can turn into a bust'em hand quite easily if you flop a straight.
• Table Dynamics: A passive table is ideal for playing this type of hand. Aggressive tables, however, warrant this one to be tossed unless you're super deep.
• Upstream Action: Unless there are excellent implied odds, toss this hand in the muck if there is any upstream action.
• How to Play:  I actually prefer QJ to KT, even though it's a 40:60 dog preflop heads up  against Katie. The reason is that it's easier to throw this one away if you get heat post-flop. It also makes straights more frequently than KT. Like all trap hands, play this one cautiously. Treat it as a drawing hand preflop and then fit-or-fold postflop if there is any action you're facing.

Queen-Ten. Quint, Cutie, Varkonyi.

• Hand Type: Very weak one gap connector.
• Position: I'll play suited QTs in MP, but only play QTo in LP. 
• Stack Sizes: Like QJ, this one is actually best suited to deeper stack play, but you absolutely cannot get married to it post-flop, too. 
• Table Dynamics: Passive tables are best, where you can see cheap flops and take down pots with cbets against weak players.
• Upstream Action: Unless there are excellent implied odds, fold this hand to any upstream action.
• How to Play:  This is the weakest of the trouble hands. You need to train your brain to inwardly wince when you're dealt a Varkonyi. In other words, play it super, super cautiously.

Jack-Ten. Justin-Timberlake.

• Hand Type: Moderate connector with semi-bluffing power.
• Position: I play JTs from MP, and JTo from LP.
• Stack Sizes: As a drawing hand, this one likes deep stacks.
• Table Dynamics: Passive tables are ideal, action tables not so much.
• Upstream Action: This is all about implied odds more than anything. Be cautious and make sure the IOs are there to get involved when facing upstream action.
• How to Play:  This used to be my favorite hand, but now I realize just how weak it is by itself. On the other hand, JT has a lot of power as a drawing hand; if you make a straight, and the board doesn't contain one of your cards, you will have the best possible straight. Treat it cautiously like the drawing hand it is, with a fit-or-fold post-flop approach, and you can't really go too wrong.

Okay, and now here's the corresponding part of the starting hand chart for these non-Ace Broadways (click on the image to enlarge if needed):

All-in for now...
-Bug

AQs Cash Hand Quiz Question

I've been fan of the Daily Quiz of the Day website for years. On average, they post more tournament than cash game questions, so as I prep for next year's WSOP I will probably be spending even more time on their web page every morning than I usually do. (You can find a link to the site in the "Resources" tab at the top of this blog's home page.)

Anyway, today's question was a fairly standard cash game Q, but it got me thinking afterward that this is a common hand that a lot of newbies lose unnecessary money on, so it bears a little scrutiny. Here's today's question via a cut and paste from their website:

So what do you do? A beginner (or even an intermediate on autopilot) might see what looks like a strongish hand, then glance at the low effective stack size and just get it in. An advanced player would take the time to do REDi:

Read: what range do we put our opponent on? He’s been laggy a lot preflop, but for 5x size raises. Here he jumps in with a 10x raise from UTG. We don’t have to worry about anyone else in the hand, as we’re in the BB. The 10x raise is weird; either it’s a very strong hand, or one he doesn’t want to see any action with. In low-stakes games I frequent, the latter is often indicative of hands like JJ or TT from a bad player. Sometimes you see people turn up AA, KK, or AK here too.

Estimate: against the stronger part of villain's range, our AQs is 2:1 dog or worse. Against a range of middle pairs, we’re at best a coin flip. We have essentially no fold equity, as he’ll be getting good pot odds to call a shove with his short remaining stack. The only real thing working in your favor is that the SPR is low; in other words, you don’t stand to lose too much with the hand.

Decide: Our hand doesn’t have much if any Value against the opps range, and we essentially don’t have any fold equity, so Semibluffing is no good. In other words, pushing is out. Calling to see a flop is not much better, as we’ll be OOP and essentially in the same boat we’re in now (i.e., confused as to where we stand). Therefore this is an easy fold.

Implement: Muck.

All-in for now...
-Bug