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.

9 Responses

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  1. Ted said, on February 28, 2010 at 8:04 pm

    Thank you for doing this. Math is great, but real world info is too. How are you running these days and how big of swings have you experienced? It’s getting harder and harder to find a $5 game around where I live and I fear a $10 minimum will be too much for my bankroll (even without Trips).

  2. stephenhow said, on March 1, 2010 at 7:54 am

    I’ve found that it all averages out okay over the long run to the -2.3% EV. I’ve also found that I can run good for over a month, playing 3 times a week, and accumulating bankroll. Then, over five bad sessions, I can lose it all back. This weekend I played 3 times and won a total of $400. But the weekend before, I lost $450 in my only session. Of course, this is all $5 Ante play, and occasionally betting Trips.

    On the other hand, I see most all other players steadily lose money by not betting their hands. Yesterday, in 3.5 hours, half a dozen people went bust this way. They’d bet $10-$15 Antes, and not raise preflop with good hands. They’d slowly bust out and leave, often leaving me heads-up with the dealer as the sole survivor, until new players showed up.

  3. Eric Z said, on March 7, 2010 at 4:28 am

    Hey – great graphs.
    Is there an error on the 2nd graph – the graph for 20 antes? You say there is a 33% for a bust, 27% for a double up, and 50% in the middle.
    That adds up to 110%.

    Should it be 40% of the results put you in the middle?

    • stephenhow said, on March 7, 2010 at 7:43 am

      Thanks, good catch. Yes, it’s 40%.

  4. Julio said, on January 4, 2011 at 6:15 pm

    What a fatalistic analysis on UTH. I have player this game for about 5-6 years and have never learned so much for one web site. Here are some “dumb” questions on the charts in this writing.

    (1) The vertical axis on the chart is said to be probabilities. Let’s use the first chart. How does “lower than 0.0005 and close to 0″ be said to be 1.5% for bustouts?

    (2) For chart 2, the text says, “… a 40% chance of falling somewhere in-between”, and the chart shows 50% in-between. Are there integrations involved? How to get 33% bustouts in the chart when the vertical axis is below 0.0005 (0.05%) and close to 0? Perhaps the arrow is pointing at the right spot?

    (3) In the chart for 200 Ante Bankroll, Leave if win 20 Antes….”, 0.2% is 0.002. The chart shows 0.002 is lower than 0.0002. Why?

    I am the first time reader, sorry in advance to ask these questions. Thanks for openning up my eyes. Julio

    • stephenhow said, on January 5, 2011 at 6:16 pm

      Julio,

      These charts show the distribution of outcomes once you’ve removed the bust-out and double-up events. In other words, they show the distribution of possibilities at the end of the last hand (you’ve reached the max hands limit). I excluded the bust-out and double-up endpoints from the graph, because they’re big spikes that swamp out the detailed distribution in-between. I replaced those big spikes with annotations like “1.5% bustout” and “69% goal”. In other words, I just chose the x-axis limits to exclude these spikes for the benefit of showing more detail in the graph.

  5. UTH fan said, on January 31, 2013 at 12:58 pm

    This is very interesting, again I wonder if playing headsup vs the dealer is different odds that playing at a full table (as I suggested this being an option in the free game). It’s easier to keep track of odds on the free website UTH games, where i find that if I can win 50% of hands and only lose less than 30% of hands (folding being very important), then I can play for a long time. While I may start out cold or hot, at 100-200 hands, the win% will settle to 50 percentish. The difference to me is how many times I lose a hand rather than fold. The biggest bust out events are losing on several consecutive 4X raising circumstances. Thus, the key is to know when to fold and not to be too unlucky that you’re only winning 40% or less of your hands. Again, I have not done the math like you have but it also feels to me that in the real world, with many players at the table, my win rate is about 33%. What win/fold/lose percentage do you find when playing will keep you afloat?

  6. dotnetchris said, on June 2, 2013 at 5:07 pm

    Stephen, you have a great blog here. It’s clear you’ve put a ton of work into it. Thank you!

  7. joe roll said, on February 13, 2016 at 1:17 am

    hi there
    im wondering about the odds if you can know all the players hole cards when playing 6-7 players
    you can discard a few large raise hands and sometimes play different on the flop or single bet
    did you ever check it?


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