Discount Gambling

Session Outcome Distributions for Ultimate Texas Hold’Em

Posted in ultimate texas hold'em by stephenhow on February 26, 2010

The most common mistake players make in UTH is to not bet their hands, especially preflop (i.e., raising 4x according to basic strategy). Certainly, if your initial bankroll is small, like 20 Antes, it’s clear that if you quickly lose three 4x raises, then you’re out. On the other hand, you can just as easily double up for a nice hit-and-run.

I’ve calculated the probability distributions for a few session scenarios, given 1) starting bankroll, 2) hit-and-run goal, and 3) maximum number of hands (session length). These distributions are calculated using basic strategy, i.e. betting a hand whenever long-term advantageous. Hopefully, these scenarios will help you understand the possible outcomes of your sessions, and their likelihoods.

100 Ante Bankroll, Leave if Double-Up or Bust-Out, 100 Hands Max

Let’s say you sit down with a bankroll of 100 Antes (i.e. \$500 for a \$5 Ante game). You decide before hand that you’ll play a maximum of 100 hands (about 3 hours), and that you’ll leave if you double up (+\$500 profit), or bust out. In this scenario, you have an equal chance (1.5%) of busting out or doubling up. Otherwise, 97% of the time you’ll end up somewhere in-between, following the distribution below:

Session Distribution for 100 Ante Bankroll/Goal, Max 100 hands

20 Ante Bankroll, Leave if Double-Up or Bust-Out, 100 Hands Max

In this example, say you have only 20 Antes (i.e., \$100 total bankroll for a \$5 Ante game). You plan to leave if you double-up or bust out, and to play a maximum of 100 hands (~ 3 hours). In this scenario, you’ll have a 33% chance of busting out, a 27% chance of doubling up, and a 40% chance of falling somewhere in-between, following the distribution below:

Session Distribution for 20 Ante Bankroll & Double-Up Goal

200 Ante Bankroll, Leave if Win 20 Antes or Bust-Out, 250 Hands Max

Say you have a big bankroll, are happy to hit-and-run after winning 20 Antes (i.e., \$100 in a \$5 Ante game), and are willing to grind out 250 hands. In this case, you have a 69% chance of making your \$100, an overall 70% chance of winning, and only a 0.2% chance of busting out. The remaining 30% of the time, you’ll fall somewhere in-between, according to the following distribution:

Session Distribution for 200 Ante Bankroll, 20 Ante Goal, Max 250 Hands

Playing “Forever” for 0.8 Antes/hr

Personally, my only goal is to play UTH “forever” and lose (2.3% Ante/hand)(35 hands/hr) = 0.805 Antes/hr. For a \$5 Ante game, that’s only a cost of \$4/hr. I consider that cheap entertainment, and Casino Pauma gives back more than that in player rewards and cashback. There, your only effective costs are your dealer tokes.

Still, there’s always the risk of running bad, and busting out a limited bankroll. (And the equal chance of running good and doubling your entire bankroll, or hitting a Royal Flush, and becoming a “lifetime winner” for a while). I computed an outcome distribution over 3000 hands, starting out with a 500 Ante bankroll (i.e., \$2500 for a\$5 Ante game), and adding back \$4/hr to your bankroll (either from cashback, or chalking it up to cheap entertainment). For this scenario, there’s still a 3.5% of busting out, and a 3.5% chance of doubling-up. Otherwise, 93% of the time, you’ll end up somewhere in-between according to the following distribution:

Distribution After 3000 Hands, 500 Ante Bankroll, Ignoring House Edge

Conclusions

The probability density function of the outcome of a single basic strategy UTH hand was calculated, and used to compute the session outcome distribution for a scenarios of bankrolls, goals, and session times. In each scenario, the simulated 2.3% Ante/hand house edge is seen, no matter the shape of the distribution.