No-limit Hold'em & coin flip
This is a rather long article of my dig into the analogy between NL hold'em
& coin flip
. I hope you enjoy reading it. Comments & corrections are most welcome.
I've heard that people using the analogy that poker is like a coin flip
. For an average player the coin is exactly 50-50. If you are highly skilled, your coin is slightly favored to your side. A 60-40 advantage would be huge
. At that time I thought "60-40 is not huge. I can still lose 40% of the time. A 90-10 would be huge". Yeah in my dreams.
For a few days during my walk to work I've been thinking of a model to describe this coin flip analogy. Let's say each hand is a coin flip. How big an advantage does a good player have on his coin
? How good is your coin?
First, let's define the problem a little. We probably agree that a good player earns, in average, about 10xBB per 100 hand
. When I played NL2 I actually did better than that (around 50XBB per 100 hands) but we all know how laughable NL2 is.
Let's use 10XBB / 100 hands as an example.
Next we need to know the average pot size. I went to PokerStars.com
and took a look at all available NL Hold'em tables at the time:
Table (BB=) Average Pot (XBB)
You can see that at higher limit, the average pot size is around 12XBB
, while this ratio grows at lower micro limits. Let's use 12 times big blind as the magic number for now.
So how does 10XBB / 100 hands look like in a coin flip?
In 100 hands, there will be a total pot
of 100 x 12BB = 1200 BB.
You get a profit of 10 BB
Let's say you are doing a coin flip of 60-40.
Per 100 average hands your net win
is 20 pots, which is 20 x 12 BB = 240 BB.
This is far better than the reality, 10BB. 24 times better.
Your real coin is 24 times weaker
than a 60-40 coin.
To be exact, it gives you an advantage of 5/6 (0.83) of the pot every 100 pots.
Your coin is a 50.417 - 49.583
That's not really an excellent coin flip you might say! But keep in mind that you are also fighting against the casino rake. From my experience, say if I have an net income of $1, the caisno has raked away $0.6 from it. This means that you are actually flipping something close to 50.7 - 49.3.
Also, most of the times you are not really participating in a pot. When you do you probably wins 60% of the time or even more. But that's exactly why you are earning money. This coin flips applies to every hand, playing or not. If you play more hands you can expect that edge to disappear or go to the red side.
Now if you look at my case. When I play NL2 I earn 50XBB / 100 hands.
In NL2 the average pot size is 25xBB.
My coin was a 51 - 49 coin at NL2. I am not really that much better than NL2 donks!
Another interesting question is to see how good the pros are doing. Last time I read an article saying that a pro is earning an hourly rate of $100
(just as an example) but his standard deviation is $600
. Now how good is his coin flip?
First I checked pokerstars and worked out the average hands / hour. It is about 90. For simplicity we will say 100 hands / hour.
Let's first simulate what is going to happen with a 60-40 coin, flipping 100 times.
If everything goes averagely, the pro wins 60 pots and loses 40 pots, netting a 20 pots average (per hour).
His standard deviation would be 0.98 pots (thanks to MS Excel).
That's better than 20:1 ratio between profit and standard deviation -- too good to be true. He is actually getting 1:6 !
The pro is actually getting a coin that is 120 times worse than the 60:40 coin. It is 50.083 vs. 49.917.
A tiny edge makes all the difference for someone to make a living out of it.
Now let's go back to my 51:49 coin. I would have an average earning of 2 pots / 100 hands, and a STDEV of 1.0 pots.
Now let's go back to your 50.4 - 49.6 coin flips.
You would earn 10BB/100 hands, which is roughly 5/6 pots per 100 hands.
Your STDEV is 1.0 pots. That's a 5:6 ratio.
You probably have seen noticed something - no matter how much your edge is, your standard deviation is always around 1.0 pots per 100 hands
. You can call this "LUCK
I still remember the first night I learned poker as a loose passive donk. After watching people bad beat each other like drinking water, I thought "poker is not really all about skill. I might say it is 3 skill -- 7 luck
in any given night".
For this 3 skill - 7 luck (3:7 ratio) or better to happen, say 100 hands are dealt during the night, you need to at least have a 50.2 vs. 49.8 coin
. Are you this good? Hopefully.
By the way, Casino games usually feature odds
much better than that. For example, a single 0 Roulette
gives the house an edge of 1/37. This is like a 0.514 vs. 0.486
coin. Once I read that a good pro's edge at a casino NL hold'em table is as high as 5% (52.5 vs. 47.5
coin). I wonder whether that only belongs to the past as average people play hold'em much better than before.
Thanks for reading. Now if you want to learn 3 things from your last 5 minutes I wish it to be:
(1) In pokers, you are usually flipping a coin (a lot) worse than 51:49 per hand.
(2) Small advantages build up to give you a profit, in the long run.
(3) Playing against the casino is worse than playing against the best poker pros.