Math disadvantage, how to calculate it?

Supose you donnot know that French roulette has 2,7% over the player and you start playing 6 numbers.

When you have played 1000 spins you are losing 10% of the total wagged(600).

After 2000 spins played you are 8% down(960).

At spin 10k you are 5% down(3000)

At 50k you are 2,7% down(8100)

We finally reach the -2,7% at 50k.

Supose the oposite, we play 6 or 20 numbers and we are 7 to 10% up most of the time until we drop to -2,7%

Is there a way to get the -2,7% from a start of +10%?

Another scenario, supose you don´t know but a wheel section is not hitting because it has a problem. You, by chance, are not playing the section, you start winning.
How do you calculate the real advantage you have when you do not play the section?

There´s many situations where you need huge data to see the real math, in between you get confused. These questions aim to clarify the math when data is not enough.

Calculating "disadvantage" requires you to make some assumptions; usually, the first one is, the wheel / dice / etc. is "fair" (i.e. each number on the wheel has the same chance of coming up as every other number; each side of a die has a 1/6 chance of coming up on any particular roll). You can't "get to" an expected result from a particular point other than theoretically. A "Monte Carlo" method (base it on a large number of spins/rolls) will give you an approximation, but there will always be some variance.

In your first example, you may never "get to" -2.7%.

As for the example with one section having a problem, without knowing exactly what the problem is and what the probabilities of each number coming up are because of it, you're stuck with taking your chances with a Monte Carlo method.

Quote: ybot

Supose you donnot know that French roulette has 2,7% over the player and you start playing 6 numbers.

When you have played 1000 spins you are losing 10% of the total wagged(600).

37 slot wheel.
In reality, The first time you were paid for a winning number you should have been able to calculate the house edge.

Empirical data, data from many actual spins, is not needed to calculate the HE when the sample space and casino payoff is known.

There is no reason not to know the HE after the first win when those 2 values are known.
The key is figuring out the sample space.

How much did the winning number pay?
You would have noticed the dealer payed you 35units to your 1unit winning wager. You got 36 units back including the winning bet.
Casino payoff is 35 to 1

Define the sample space. This should be easy to understand.
{0,1,2,3,...,36} 37 possible winning numbers.
Probability of one number winning is 1/37
(favorable outcomes / total possible outcomes from the sample space)

Knowing the sample space we can figure out the true (fair) odds
True odds = 36 to 1 (36 ways to lose TO 1 way to win)
you should have received 37 units back including your winning wager. 36 + 1

They, the casino, shorted you 1unit in the payoff.
Any 10 year old could see this.
1unit is The difference between the True Payoff and the Casino actual payoff.
You should have received 37 units total.

HE = difference / true total payoff or 1/37

0.027027... is just the decimal equivalent for the fraction 1/37
(0.027027... = 27/999)

learn to define the sample space.
If you can not, you are then left to deal with Empirical data or formulas.
It makes the math so much easier to calculate HE.

This is just one way to calculate the HE based on what data you can gather.
if you know the fair (true) payout and the actual payout: (f-a)/(f+bet),
where f=fair payout and a=actual payout.

We know the -2,7%.

We are trying to predict fluctuacions and the coming to the real advantage.

Quote: ybot

When you have played 1000 spins you are losing 10% of the total wagged(600).

After 2000 spins played you are 8% down(960).

At spin 10k you are 5% down(3000)

At 50k you are 2,7% down(8100)

We finally reach the -2,7% at 50k.

OK.
So, what happens when you reach -2.7% at 2,458 spins?
And 3.5% at 3,682 spins?

This is variance.
What are you exactly trying to calculate?
Do you understand the difference between the variance of the wager and variance of the number of wins?
They are two different things.

I know several things about standard deviations and variance.

My questions aim to find a sort of formulae to estimate the -2,7% when you have played 1000 2000 or 5000 spins and you are not -2,7%, you are -5, -10 or +10%.

Quote: ybot

I know several things about standard deviations and variance.

My questions aim to find a sort of formulae to estimate the -2,7% when you have played 1000 2000 or 5000 spins and you are not -2,7%, you are -5, -10 or +10%.

OK that is good.
Review of some basics. (no proofs offered. I may link to those later or you can find the proofs in a probability book)

Euro Roulette
one number bet.
parameters
1/37 = p
36/37 = q (1-p)
1000 spins = n

expected value (mean) = n*p = (1000/37)
binomial standard deviation = square root(N*P*q) [(N*P*q)=variance]
square root(1000*1/37*36/37) = 1SD

expectation
= (1000/37) mean +/- 1SD
You can do the math.(I hope you got ~27.03mean and ~5.1 for 1SD)
wikipedia shows the probabilities of what 1 SD is.
1SD range is 21.9 to 27.1
This is what the central limit theorum is all about.

Now, This has not answered your question, but lays out the process that is used.
Let us deal with a house edge formula and a formula for the variance of a wager.

For a wager that pays x:1 with a probability of winning p, the house edge and standard
deviation (per unit bet and per 'root decision') are:
x= payoff (x to 1)
1-p = q we already know this.

he = (x+1)*p - 1 
std = (x+1)*Sqrt[p*(1-p)]
You can do the math. The SD value is for a 1 unit bet only.

***Let us know what values you have found.

Example: This is your assignment
parameters:
1/37 = p
36/37 = q (1-p)
1000 spins = n
1unit wager ($1)
payoff 35:1

Calculate the expected value for 1000 bets.
= he * bet * n (maybe 1/37*$1*1000)

The SD of Xnumber of bets is the square root of n (the number of trials) times the SD for 1.
That value you need to calculate from the above formula.std = (x+1)*Sqrt[p*(1-p)]
This is very important to understand.

Once you have this down then we can compare actual to expected to see how far away one is and that probability.

But you need to understand and be able to do the math for the example above or the last step can not be take by me.

I will be back tonight for your answers.

This has been done before in other threads. I may also find a few and link to them.
If I show you all the numbers now, you may not be able to do the math on your own.

If you just want numbers and do not care to know how to do it yourself, I will give them to you later tonight.

I know how to calculate SDs and the mean.

I know the 68-95-99,7 rule too.

Let´s see the question from another side, supose we play on a 12 number-section of the wheel that has a defect that increase the HE to 10% but we don´t now it yet.

We´ll find it out after playing 30k spins(late).

Is there any way to know it beforehand?

Quote: ybot

Let´s see the question from another side, supose we play on a 12 number-section of the wheel that has a defect that increase the HE to 10% but we don´t now it yet.

We´ll find it out after playing 30k spins(late).

Is there any way to know it beforehand?

Like (I thought) I said before: No - unless you are able to determine the HE before the first spin.

Let me change the problem slightly: suppose the defect in the wheel builds up with each spin, so it is 6% when you start, but 7% after 1000 spins, 8% after 2000 spins, and 9% after 3000 spins. How can you possibly determine this in advance based solely on what numbers come up?

How many spins do you think you need in order to calculate the "current" HE?
Keep in mind that, in roulette, if the HE for a particular bet increases, the HE for "betting against it" decreases. For example, if the HE for Red is 10%, then Red comes up 5/11 of the time, so "Black and Green" come up 6/11 of the time; on a European wheel, bet 18 on Black and 1 on 0 - 5/11 of the time, you lose 19, but 6/11 of the time, you win 18, so the HE is (95/11 - 108/11) / 19 = -6.22%.
(And why do I have the feeling that the original post was really asking, "How can I detect the true HE so my system will work"?)

Another view to understand(me) the situation.

Supose I know I have an edge over the bank but I donnot know the real one.

It might be 2 3 5 or 10% but at the beginning of the game I see that my edge is 10%, then 12%, after 3k 6%, and so, due to fluctuations.

As we donnot know the mean it is hard to calculate the SDs to estimate fluctuations.

How would I calculate it?

Quote: ybot

Let´s see the question from another side, supose we play on a 12 number-section of the wheel that has a defect that increase the HE to 10% but we don´t now it yet.

We´ll find it out after playing 30k spins(late).

Is there any way to know it beforehand?

No. Not from HE only.
There is a way to calculate the HE after N trials but that assumes equal bets. And then you still have to deal with confidence intervals.
I do not think it could find bias.

Wheel bias, as I understand from reading Stewart N. Ethier papers and books on the subject, is about relative frequency and the Chi-squared test and other confidence interval formulas, none dealing with HE after n trials.


Those that believe that "hot" numbers and "sleeping" numbers are an extreme rare event and can be taken advantage of do not understand the law of large numbers and the central limit theorem and their applications.
Many fallacies come from misunderstanding LLN and CLT.

Divide your advantage squared by your shoe size

We are looking for a way to know something beforehand.
The example of a defective section is good to show the scene.
You think that you have -2.7% but the real HE when you play on the section is 10%.

Is there a way to know it after 2000 trials?

See the answer above. If you do not understand it, seek out a Professor Emeritus of Mathematics