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.)