Doc
Doc
Joined: Feb 27, 2010
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September 28th, 2011 at 9:55:38 PM permalink
Quote: statman

... As before, 00 is the biased number with a probability of 3/76 at the expense of No. 36, which is given a probability of 1/76. All other numbers have a probability of 1/38. ...


If you have a wheel that is so outrageously biased that one number is 50% more likely to be hit than the other numbers ?????

I think this is something similar to having the slot for number 00 being 50% wider than the average slot. People would look at the wheel and say, "WTF is going on with the slot for 00?" If a wheel is THAT bad, people can probably figure it out without benefit of these marvelous tables.

Now if a wheel actually does have bias but not something so absurd as in this example, yes, I suspect it will probably take thousands of spins to establish that it is biased and by how much. Maybe I'm wrong. Suppose some shoddy casino has a really wobbly wheel that gives 1/2% or 1% bias toward a number. Which of your tables would pick that up and how many spins would it take for you to be sure?
statman
statman
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September 28th, 2011 at 11:13:27 PM permalink
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guido111
guido111
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September 28th, 2011 at 11:20:17 PM permalink
Quote: statman

"You can directly calculate F(N,M) = probability that at least one of 38 equally-likely outcomes appears N or more times in M trials."

Yes, that is precisely what I do, and that is what goes into the tables, but I haven't seen anybody else do it. I think you understand the problem now and if you stick at it you may come up with the answer. Here's a hint: "inclusion-exclusion principle."


My 2 cents after seeing statman's tables that he kindly sent to me.

I was unable to exactly match some of his data.
So I start here.
Nope27 Blog

http://wizardofvegas.com/member/nope27/blog/

"Roulette 00 Wheel "at least 1" number NOT hitting
September 26th, 2010 at 11:21:44 am

The below table is for the probability of NOT hitting "at least 1" number in x spins

formula used from:http://wizardofodds.com/askthewizard/roulette.html
Wizard of Odds article

question #2 at WoO. (The Wizard also shows results from his computer simulation)

Wizards' Example: for "at least 1" number NOT hitting in 200 spins
Sum i=1 to 37 [(-1)^(i+1) × combin(38,i) × ((38-i)/38)^38] = 16.9845715651245%"

This is actually easy to set up in Excel and I verified The Wizard's and Nope27's results exactly for 200 spins.

.16794 statman results.
added: snapshot of statman's table

We should all return exact results no matter what correct formula we use. Yes?
My quick 1 million sim results shows 0.169984

group middle freq freq/100
----------------------------------------------
33.50 <= x < 34.50 34.00 7 0.00%
34.50 <= x < 35.50 35.00 510 0.05%
35.50 <= x < 36.50 36.00 12311 1.23%
36.50 <= x < 37.50 37.00 157156 15.72%
37.50 <= x < 38.50 38.00 830016 83.00%

----------------------------------------------
34.00 7
35.00 510
36.00 12311
37.00 157156
38.00 830016
----------------------------------------------
cumulative
----------------------------------------------
33.50 <= x < 34.50 34.00 7 0.00%
34.50 <= x < 35.50 35.00 517 0.05%
35.50 <= x < 36.50 36.00 12828 1.28%
36.50 <= x < 37.50 37.00 169984 17.00%
37.50 <= x < 38.50 38.00 1000000 100.00%

----------------------------------------------
34.00 7
35.00 517
36.00 12828
37.00 169984
38.00 1000000
grouped data
items: 1000000

minimum value: 34.00
first quartile: 38.00
median: 38.00
third quartile: 38.00
maximum value: 38.00

mean value: 37.82
midrange: 36.00

range: 4.00
interquartile range: 0.00
mean abs deviation: 0.30

sample variance (n): 0.18
sample variance (n-1): 0.18
sample std dev (n): 0.42
sample std dev (n-1): 0.42

Continue please...

"at least 1" number NOT hitting in 150 spins
statman 0.50455
theoritical 0.517748
my simulation (1 million sessions) 0.517692

group middle freq freq/100
----------------------------------------------
30.50 <= x < 31.50 31.00 1 0.00%
31.50 <= x < 32.50 32.00 13 0.00%
32.50 <= x < 33.50 33.00 234 0.02%
33.50 <= x < 34.50 34.00 2848 0.28%
34.50 <= x < 35.50 35.00 23314 2.33%
35.50 <= x < 36.50 36.00 122079 12.21%
36.50 <= x < 37.50 37.00 369203 36.92%
37.50 <= x < 38.50 38.00 482308 48.23%

----------------------------------------------
31.00 1
32.00 13
33.00 234
34.00 2848
35.00 23314
36.00 122079
37.00 369203
38.00 482308
----------------------------------------------
cumulative
----------------------------------------------
30.50 <= x < 31.50 31.00 1 0.00%
31.50 <= x < 32.50 32.00 14 0.00%
32.50 <= x < 33.50 33.00 248 0.02%
33.50 <= x < 34.50 34.00 3096 0.31%
34.50 <= x < 35.50 35.00 26410 2.64%
35.50 <= x < 36.50 36.00 148489 14.85%
36.50 <= x < 37.50 37.00 517692 51.77%
37.50 <= x < 38.50 38.00 1000000 100.00%

----------------------------------------------
31.00 1
32.00 14
33.00 248
34.00 3096
35.00 26410
36.00 148489
37.00 517692
38.00 1000000
grouped data
items: 1000000

minimum value: 31.00
first quartile: 37.00
median: 37.00
third quartile: 38.00
maximum value: 38.00

mean value: 37.30
midrange: 34.50

range: 7.00
interquartile range: 1.00
mean abs deviation: 0.67

sample variance (n): 0.63
sample variance (n-1): 0.63
sample std dev (n): 0.80
sample std dev (n-1): 0.80

"at least 1" number NOT hitting in 100 spins
statman 0.93518
theoritical 0.953397
my simulation (1 million sessions) 0.953301

group middle freq freq/100
----------------------------------------------
26.50 <= x < 27.50 27.00 2 0.00%
27.50 <= x < 28.50 28.00 8 0.00%
28.50 <= x < 29.50 29.00 149 0.01%
29.50 <= x < 30.50 30.00 925 0.09%
30.50 <= x < 31.50 31.00 5115 0.51%
31.50 <= x < 32.50 32.00 21848 2.18%
32.50 <= x < 33.50 33.00 69154 6.92%
33.50 <= x < 34.50 34.00 159213 15.92%
34.50 <= x < 35.50 35.00 256162 25.62%
35.50 <= x < 36.50 36.00 271879 27.19%
36.50 <= x < 37.50 37.00 168846 16.88%
37.50 <= x < 38.50 38.00 46699 4.67%

----------------------------------------------
27.00 2
28.00 8
29.00 149
30.00 925
31.00 5115
32.00 21848
33.00 69154
34.00 159213
35.00 256162
36.00 271879
37.00 168846
38.00 46699
----------------------------------------------
cumulative
----------------------------------------------
26.50 <= x < 27.50 27.00 2 0.00%
27.50 <= x < 28.50 28.00 10 0.00%
28.50 <= x < 29.50 29.00 159 0.02%
29.50 <= x < 30.50 30.00 1084 0.11%
30.50 <= x < 31.50 31.00 6199 0.62%
31.50 <= x < 32.50 32.00 28047 2.80%
32.50 <= x < 33.50 33.00 97201 9.72%
33.50 <= x < 34.50 34.00 256414 25.64%
34.50 <= x < 35.50 35.00 512576 51.26%
35.50 <= x < 36.50 36.00 784455 78.45%
36.50 <= x < 37.50 37.00 953301 95.33%
37.50 <= x < 38.50 38.00 1000000 100.00%

----------------------------------------------
27.00 2
28.00 10
29.00 159
30.00 1084
31.00 6199
32.00 28047
33.00 97201
34.00 256414
35.00 512576
36.00 784455
37.00 953301
38.00 1000000
grouped data
items: 1000000

minimum value: 27.00
first quartile: 34.00
median: 35.00
third quartile: 36.00
maximum value: 38.00

mean value: 35.36
midrange: 32.50

range: 11.00
interquartile range: 2.00
mean abs deviation: 1.15

sample variance (n): 1.98
sample variance (n-1): 1.98
sample std dev (n): 1.41
sample std dev (n-1): 1.41

I do not have or have seen statmans formula for calculating at least 1 or more in N spins. I can only run simulations at this point in time.
150 spins at least 1# with exactly 9 hits
.371208 sim results
.36028 statman results.

150 spins at least 1# with exactly 5 hits exactly
.998709 sim results
.99843 statman results.

We are getting closer but I have a feeling that a formula(s) somewhere is in error.

Probably with PIE (principle of inclusion-exclusion) I love chocolate pie.

This is a good math exercise for any that want the challenge.
Statman did a good job for what I have seen so far.
I just could not verify his results at this time.
statman
statman
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September 28th, 2011 at 11:54:35 PM permalink
MATERIAL REMOVED

Some of it eventually may reappear on the web site of the
Rancocas Valley Journal of Applied Mathematics
Please flag this page so that it may be deleted.
Many thanks to those who have been helpful.
A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant
statman
statman
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September 29th, 2011 at 2:00:43 AM permalink
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Please flag this page so that it may be deleted.
Many thanks to those who have been helpful.
A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant
DJTeddyBear
DJTeddyBear
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September 29th, 2011 at 5:31:07 AM permalink
Quote: statman

Here is a message addressed directly to the Wizard on his private e-mail.

Dear Shack,

I would like to draw your attention to the thread "New Probability Tables for Roulette IV" in the Free Speech Zone of your forum and recommend that you expel buzzpaff from the forums. He is a relatively new member and has shown himself to be an ignorant and malicious bully.

If, having made a thorough study of the material presented there, you agree with buzzpaff, please delete the entire thread..

Buzz is a new member?

buzzpaff
Member since: Mar 8, 2011
Threads: 56
Posts: 1441

statman
Member since: Sep 25, 2011
Threads: 5
Posts: 27

You're here four days and you've caused more ruckus than Buzz has in his 6.5 months!
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁 Note that the same could be said for Religion. I.E. Religion is nothing more than organized superstition. 🤗
DorothyGale
DorothyGale
Joined: Nov 23, 2009
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September 29th, 2011 at 6:56:00 AM permalink
Quote: statman

The Pr that at least one of the numbers 0 through 37 will not be hit by the 200th spin can be expressed
as 1 - (1 - (37/38)^200)^38. This evaluates to 0.16793785. The Wizard gets 0.1698457.



Simulation of 10M experiments where 200 spins are performed in each experiment ...

Percentage of times 1 or more numbers un-hit: 0.169852


#include <stdio.h>
#include <time.h>
main() {
int c1, c2, i;
int hits[38] = {}; // 37 = 00
double r = 0;
srand(time(NULL));
for (c1 = 0; c1 < 10000000; c1++) {
i = 0;
for (c2 = 0; c2 < 38; c2++)
hits[c2] = 0;
for (c2 = 0; c2 < 200; c2++)
hits[rand()%38]++;
for (c2 = 0; c2 < 38; c2++) {
if (hits[c2] == 0) {
i = 1;
break;
}
}
if (i == 1)
r++;
}
printf("Percentage of times 1 or more numbers un-hit: %1.6f\n", r/10000000);
}


This code is easy enough to write, even in Fortran, that one wonders why Statman did not check his results by a larger simulation.

Edit: Since I posted this, just moments ago, Statman has edited all of his posts saying "goodbye" ... I assume he is embarrassed at how wrong he was after the guys & gals looked at this work ... because he is gone, let me editorialize and say he was one of the most arrogant and unpleasant voices on this site in quite a while. His "know it all" pedestal grew very tiring ... always talking down to those who clearly have more expertise, education and insight than he does ... thanks Guido for taking the time and skill to open the floodgate ...

--Ms. D.
"Who would have thought a good little girl like you could destroy my beautiful wickedness!"
DJTeddyBear
DJTeddyBear
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September 29th, 2011 at 8:30:05 AM permalink
Quote: DJTeddyBear

Buzz is a new member?

...

You're here four days and you've caused more ruckus than Buzz has in his 6.5 months!

FYI: In a private message, statman took exception to these comments. I believe my reply didn't please him, prompting him to remove his posts.

FWIW: I think it was a mistake to remove the posts.
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁 Note that the same could be said for Religion. I.E. Religion is nothing more than organized superstition. 🤗
DorothyGale
DorothyGale
Joined: Nov 23, 2009
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September 29th, 2011 at 9:21:57 AM permalink
Quote: DJTeddyBear

FYI: In a private message, statman took exception to these comments. I believe my reply didn't please him, prompting him to remove his posts.

I don't understand why he used you as an excuse to leave ... he didn't use you as an excuse to arrive ... silly, childish & irrational person ...

I got the bug to finish the code I wrote ... for the record:


#include <stdio.h>
#include <time.h>
main() {
int c1, c2, x;
int hits[38] = {}; // 37 = 00
double r[38] = {};
double c[38] = {};
srand(time(NULL));
for (c1 = 0; c1 < 10000000; c1++) {
x = 0;
for (c2 = 0; c2 < 38; c2++)
hits[c2] = 0;
for (c2 = 0; c2 < 200; c2++)
hits[rand()%38]++;
for (c2 = 0; c2 < 38; c2++) {
if (hits[c2] == 0) {
x++;
}
}
r[x]++;
}
for (c1 = 0; c1 < 38; c1++)
for (c2 = c1; c2 < 38; c2++)
c[c1] += r[c2];

printf("10 million trials of 200 spins ...\n");
printf("----------------------------------\n");
for (c2 = 0; c2 < 38; c2++)
printf("%2d %1.6f %1.6f\n", c2, r[c2]/10000000, c[c2]/10000000);
}



Output -- 10M rounds of 200 spins each, with a bit of editing to make the output clearer ...



p = probability of not hitting that exact number in 200 spins
q = probability of not hitting that number or more in 200 spins
--------------------------------------
# not hit p q
0 0.830074 1.000000
1 0.156806 0.169926
2 0.012547 0.013119
3 0.000557 0.000572
4 0.000015 0.000015
5 0.000001 0.000001
6 0.000000 0.000000

"Who would have thought a good little girl like you could destroy my beautiful wickedness!"
statman
statman
Joined: Sep 25, 2011
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September 29th, 2011 at 10:56:05 AM permalink
MATERIAL REMOVED

Some of it eventually may reappear on the web site of the
Rancocas Valley Journal of Applied Mathematics
Please flag this page so that it may be deleted.
Many thanks to those who have been helpful.
A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant

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