Here is result of 120 spins. How to calculate standard deviation across the best 7 numbers to play?
| 0 | 3 |
|---|---|
| 1 | 6 |
| 2 | 3 |
| 3 | 5 |
| 4 | 1 |
| 5 | 3 |
| 6 | 3 |
| 7 | 1 |
| 8 | 6 |
| 9 | 6 |
| 10 | 4 |
| 11 | 7 |
| 12 | 5 |
| 13 | 6 |
| 14 | 5 |
| 15 | 6 |
| 16 | 4 |
| 17 | 3 |
| 18 | 4 |
| 19 | 6 |
| 20 | 3 |
| 21 | 3 |
| 22 | 3 |
| 23 | 5 |
| 24 | 2 |
| 25 | 2 |
| 26 | 1 |
| 27 | 1 |
| 28 | 1 |
| 29 | 3 |
| 30 | 2 |
| 31 | 1 |
| 32 | 0 |
| 33 | 1 |
| 34 | 1 |
| 35 | 3 |
| 36 | 1 |
To answer this question I will first calculate SD for each number hit.
From that we get:
Probability to win for a single number on roulette would be 1/37
Probability to lose for a single number on roulette would be 36/37 or (1 - Probability to win = 1 - 1/36 )
Trails is 120 spins.
From that we get
SD = SQRT (1/36 x 36/37 x 120)
SD = SQRT (3.15)
SD = 1.77
The Mean, (expected hits on each number for even distribution) for 120 spins on 37 numbers is 120/37 = 3.24
Let’s look from our data position the highest hit rate position 11.
It has 7 hits.
Now let’s calculate how it lies from the mean?
It is called Z-score
z = ( 7 – Mean) / SD
z = (7 – 3.24) / 1.77
z = 2.12
Position 11 with value 7 has z-score value of 2.12.
Values within one standard deviation of the mean have 68% chance to happen;
two standard deviations of the mean about 95%;
and within three standard deviations about 99.7%.
You can see it on this chart.
From that chart we can only approximate so here is a table with values.
How to find our z-score = 1 in the table?
Scroll down to find in blue 1 row then go left to find in green 0.00.
You should find value 0.8413
z-score can be and negative number but in the table we don’t have data for such case. You can make one but it is basically in this case 1 - 0.8413 = 0.1587. So number 1 minus whatever is on positive side.
If you look chart you can see that z-score 0 has value 0.5. Of course it is in middle and there is a 50:50 chance for each side.
If i deduct 0.1587 from 0.8413 i get
0.8413 = 0.1587 = 0.6826
which means 68% of hits will be within 1 SD.
That matches previously stated
Values within one standard deviation of the mean have 68% chance to happen;
Now if we look our z = 2.12 in the table we find 0.9826
Position 11 has highest value than 98% of expected values of the other positions.
We can look z-score for all numbers.
Now lets look sections of 7 pockets.
For that I extended data on each side by 3 pockets as on the picture.
Before 0 copied 34,35,36 and after 0 copied 0,1,2, it makes it easier to add ends of data.
Added 7 cells together and divided by 7 which gives me an average.
To find how much hits are from an average in this case 120/37=3.24
Subtract from amount of hits 3.24
Then divide result with 3.24.
Bingo
Make a chart and you get something as this.
Showing 72% advantage at 11,12
Median would be in middle of green area so 14 which could be safer to play.
So let’s look now SD across 7 pockets sector
SQRT (7/37 x 30/37 x 120) = SQRT 18.4 = 4.29
It is higher.
But Mean in this case would be (120 / 37) x 7 = 22.7
And around position 12 on 7 pockets sector we have 39 hits.
So let’s check z-score;
Z = (39 – 22.7) / 4.29 = 3.79
You can see z-score is much higher even the normal SD distribution is higher.
It tells you that almost there is no chance that so much higher amount of hits across 7 pockets is just a coincidence.
While z-score or how we like to say amount if standard deviations tells us probability we can easy calculate our possible winnings based on the ball jumps data.
In our example we have 39 hits on the sector of 7 pockets in 120 spins we played.
If we play 7 numbers sector for 120 spins we would play;
7 x 120 = 840 units
We would win 36 units 39 times therefore
36 x 39 = 1404
And profit 1404 – 840 = 564 units
If we divide profit with units we invested we get advantage
564 / 840 = 0.67
It is 67% advantage
It means for each $100 we place on the table we would profit $68
We actually changed the casinos – 2.7% advantage, so the total change is about 70%.
On same principle we can calculate casinos advantage for the particular bet
For a single number
Chance to win is W = 1/37
Chance to lose L= 36/37
35 is amount of units we win
W = 1/37 x 35 = 0.027 x 35 = 0.945
1 is amount of units we lose
L= 36/37 x 1 = 0.972
We subtract L form W.
Advantage = 0.945 – 0.972 = - 0.027; or – 2.7%
On a same way we can calculate it for a spit bet
On a split bet we would win 17 units, 2 times on 37 spins and lose 35 times 1 unit;
Advantage = W – L = (2/37 x 17) – (35/37 x 1) = 0.918 – 0.945 = - 0.027
Someone may get confused and say but on a split bet I win 18 units because I get my chip as well. It is truth but then you wouldn’t lose one unit 35/37 but 37/37 times and calculation would be;
Advantage = W – L = (2/37 x 18) – 37/37 = 0.972 - 1 = -0.027 = - 2.7%
Hi,
any chance of fixing the images, or share the excel with that math please?
Thank you very much
Looks like the same wheel, about which we talked in neighbour posts , i writed formula how to calculate…
I would like to know this as well
Can you link to that “neighbor” post? Or is it private?
Thank you
Almost same, but notice this is a question. When creating post there is tag area , if you write question the thread gets another format. Also members can give votes to answers they like.
You can replay to each post, or you can answer. Also the answers get sorted by highest vote.
Not sure who would be the best for this subject perhaps @Snowman.
What wheel is that data from? And what ball? Looks kind of similar to my jump chart.
I think the best is ~15
It is , just an example data.
Good writed and good will be will write source.
But here is again the same problem , that we to all look after all happend. So we have data - find best place and for it we look how much sigmas we have…
Should reply to the answer instead of making the answer but commenting it.
I am not sure, if looking only 2 numbers at any place than yes. But if looking next to each other numbers where most of them have significantly higher hits it is a different story.
If the chart looks as yours it would be close to impossible to be a coincidence, that next same amount of spins the pick gets shifted ~ ,9,18, 27 pockets. Few points in the high area may change in value but whole still should be there. For bias is different you look say 2 numbers at any place, in say 3x37 spins each one should get 3 hits but if 2 numbers get 5 it means nothing but if 0-18 mostly all are above average it already have meaning.
1 SD has its meaning. The point is that you could see how with more numbers it increasing. We already made decision to look numbers next to each other but not just any with highest hits. That makes it different from what bias player may do.
Great post Forester!
Yes you are correct
While we are at it. Could you guys give an example how to calculate the CHi square ?
Thank you
From the provided data, we can select the top 7 numbers based on their frequency or any other criteria. For this example, let’s choose the numbers 11, 9, 1, 2, 6, 13, and 14.
Calculating the Mean:
The first step is to calculate the mean (average) of these 7 selected numbers. Add them together and divide by the total count. In this case:
\text{Mean} = \frac{11 + 9 + 1 + 2 + 6 + 13 + 14}{7} \approx 7.7143Mean=
7
11+9+1+2+6+13+14
≈7.7143 (rounded to four decimal places) .
Squared Differences:
Next, we find the squared difference between each selected number and the mean. For example:
(11 - 7.7143)^2 = 12.5715(11−7.7143)
2
=12.5715
(9 - 7.7143)^2 = 1.6531(9−7.7143)
2
=1.6531
(1 - 7.7143)^2 = 46.0408(1−7.7143)
2
=46.0408
(2 - 7.7143)^2 = 32.5715(2−7.7143)
2
=32.5715
(6 - 7.7143)^2 = 2.9745(6−7.7143)
2
=2.9745
(13 - 7.7143)^2 = 28.9802(13−7.7143)
2
=28.9802
(14 - 7.7143)^2 = 39.7551(14−7.7143)
2
=39.7551
Sum of Squared Differences:
Add up all these squared differences to get the sum: 12.5715 + 1.6531 + 46.0408 + 32.5715 + 2.9745 + 28.9802 + 39.7551 = 164.566912.5715+1.6531+46.0408+32.5715+2.9745+28.9802+39.7551=164.5669
Why he must help you ? Why not oposite ?