The Math behind HU SNGs

[jordanbillie]

Hello everybody, today I did a simple math exercise to determine what my win rate must be in order to be profitable at Heads Up SNGs, specifically on BetOnline. Here are my results:

First I should state the rake for HU SNGs on BoL, which is 5.6% (the $1.50 games are $1.42 + .08). In the $1.50 games two things can happen, you either lose (-$1.50) or you win (+$1.34). I think it is obvious that if your win rate was 50/50 (50%) you would not be a profitable player, you would have an ROI of -5.3%. Here is the simple way you can calculate that with a sample of just 2 games: 1 win (+1.34) and 1 loss (-1.50) = 16 cents lost.
ROI = Profit/Amount invested (ROI = -0.16/3.00, ROI = -5.33%

So...what does our win rate have to be in order to be profitable and beat the rake?

Here are my results for some different win rates:

50% win rate = -5.3% ROI
52% win rate = -1.5% ROI
53% win rate = 0.3% ROI
55% win rate = 4.1% ROI
60% win rate = 13.6% ROI

It turns out that the break even point is winning half the rake over 50%, so in this specific case with 5.6% rake you'd have to have a win rate greater than 52.8% to be profitable (52.8169014084507% specifically).

Currently my win rate is exactly 60%, so I should just focus on sustaining that win rate and I'll be quite fine.

Jordan

 

[dayontour]

Good to see the real math

Great post, good to see the real math behind what you may think or wonder.

Thanks for posting.

 

[jordanbillie]

Also, side question: What is the best winning streak you have ever had in heads up SNGs? Mine is 8.

 

[BelFish]

Also, side question: What is the best winning streak you have ever had in heads up SNGs? Mine is 8.

17 or 18 in 1.5$ SnGs HU, but it was not NLH ))

P. S. Somewhere i posted a topic in which there is a formula for calculating the probability of streaks of different lengths depending on ROI. I'll try to find it later...

 

[Marshmalo1994]

I never tried to play HU sngs. Maybe I'll try, at least to get some fun with low BIs, and see what happens.
Thank for the tips.

 

[BelFish]

I thought the formula was in this thread, but it's in some other thread:

Longest winning streak in CC freerolls on PS

Intresting which the longest winning streak a few days in a row had somebody... Personally, i have a small streak, 3 days in a row. But i play according to my mood: sometimes it's tight, and sometimes i can put all-in with any cards at the beginning of the tournament ))) Maybe there are...

www.cardschat.com

But there in the 47th post there are some calculations using this formula. I'll try to find the formula later, but for now, here are the some calculations for winning streaks in SnGs HU:

For fun, i also calculated the probability of various winning streaks for the games in SnG HU.

If a player wins about 64% SnGs HU, then the probability of hitting a winning streak of 20 wins in a row for 3000 SnGs will be 13.3%

If he wins about 70% SnGs, then this probability will already be 51.2%, and the probability of a streak of 30 consecutive wins for 3000 SnGs in this case will be slightly less than 2%.


I managed to bring my streak, which i wrote about above, to 17 victories in a row. And the funny thing is that a full fish beat me, but damn lucky fish ))
And after playing with him, i beat 3 pretty strong players in a row. So the series could not be 17, but 21 times in a row. Such a bummer ))

 

[BelFish]

Found through the search filter for the search query "Formula". Found in this thread:

Royal-flush twice in a row. What is the probability?

We all know that the rarest hand in poker is the royal flush. Some players have not collected it for many years. Collecting a royal flush is a rarity. Collecting it twice in a row is a miracle. It succeeded. It was an American poker player - Alex. The probability of such an event is 0.00000004%...

www.cardschat.com

-----------------------

And here is that complex formula for calculating at a distance of N hands (you can write this formula in Excel cells and use it to calculate the probabilities of various streaks):

A1 is P
A2 is K
A3 is N
A6 is (p), or the probability of the event for the repeat series we are looking for.

A4=1+(1-A1)*A1^A2+(A2+1)*(1-A1)^2*A1^(2*A2)

A5=((1-A1*A4)/((A2+1-A2*A4)*(1-A1)))/A5^(A3+1)

A6=1-A5

Here is an example of calculating this formula for playing roulette: the probability that 11 or more times in a row will be one color, for example, red at a distance of 2000 games, is 47%, and for a distance of 5000 games this probability will be equal to 80%

Here we substitute K=11, P=18/37 into the formula and calculate for cases N=2000 and N=5000

K is the number of repetitions of the event in a row in a series of N games, and P is the probability of this event.

Similarly, using the same formula, you can calculate the probability of getting a royal flush 2 times in a row at any distance. To do this, you need to substitute K=2, P=0.00001 into the formula and see what happens for different values of N (1000, 10000, 100000, 10000000)

 

[rvicente13]

Wow, thanks for sharing! A personal question, what is your longest losing streak in a heads up match? Have you ever had a day that wasn't your day?

 

[BelFish]

Wow, thanks for sharing! A personal question, what is your longest losing streak in a heads up match? Have you ever had a day that wasn't your day?

Probably about 10 losses in a row. I don’t remember that there was a very long losing streak...

----------------

Have you tried writing the formula in excel?

To check whether everything is considered correct, it must match for those examples that were above. For example, for 3000 SnGs HU games with a frequency of wins of 70%, the probability of a streak of 30 wins should be equal ~ 2%.