Benford's Law

[Karnac]

I found this on the rec.gambling newsgroup......kinda interesting when applied to lottery numbers


Here is something that you may already be aware of, but if you have
not heard of Benford's Law before, you will find this of interest
(and hard to believe!!!)

Intuitively, most people assume that in a string of numbers sampled randomly
from some body of data, the first non-zero digit could be any number from
1 through 9. All nine numbers would be regarded as equally probable.
But, as Dr. Benford discovered, this is not so.

Given a string of numbers, the chance that the first digit will be 1 is not
one in
nine, as you would imagine; according to Benford's Law, it is 30.1 percent,
or
nearly one in three. The chance that the first number in the string will be
2 is
only 17.6 percent, and the probabilities that successive numbers will be the
first
digit decline smoothly up to 9, which has only a 4.6 percent chance.

So what I hear you say, well the income tax agencies of several nations and
several US states, large companies and accounting businesses are using
detection software based on Benford's Law as a powerful and relatively
simple
tool for pointing suspicion at frauds, embezzlers, tax evaders, sloppy
accountants
and even computer bugs.

So what are the probabilities of the first digits. Well it follows the
following
formula : the probability of any number "d" from 1 through 9 being the first
digit
is log to the base 10 of (1 + 1/d). i.e.

Digit Probability
1 30.10%
2 17.61%
3 12.49%
4 9.69%
5 7.92%
6 6.69%
7 5.80%
8 5.12%
9 4.58%

Hard to believe but true. We (some friends and I) have tested it on a sample
size
of 8,000, 55,000 and 65,000 numbers, and it all closely matches the above.

Further reading
http://www.rexswain.com/benford.html
http://www.math.yorku.ca/Who/Faculty/Brettler/bc_98/benford.html
http://www.mathpages.com/home/kmath302/kmath302.htm

 

[patrickim1]

Thanks Karnac

I've learned something new today...

Looks like there's something true about that... well, any idea is worth checking when trying to crack the Jackpot...

 

[PAB]

What an Excellent Site Karnac ,

http://www.mathpages.com/home/kmath302/kmath302.htm

Certainly NOT One to Quickly Look Through .
There is a Mine of Information Just Waiting to be Learnt .
BTW Karnac, I Just Noticed you Reached 1,000 Posts, Congratulations.

All the Best
PAB

 

[Karnac]

What an Excellent Site Karnac ,

http://www.mathpages.com/home/kmath302/kmath302.htm

Certainly NOT One to Quickly Look Through .
There is a Mine of Information Just Waiting to be Learnt .
BTW Karnac, I Just Noticed you Reached 1,000 Posts, Congratulations.

All the Best
PAB

Thanks PAB. Glad you liked that site ...the credit for it goes to a gentleman named Jack as he was the one who first posted it. There certainly is a huge amount of information there...I only wish I had the smarts to comprehend it....Math was a nightmare for me ....grade 9 math was the toughest 3 years of my life ....and I've never really needed it since Enjoy!

 

[GillesD]

I tried to duplicate those results, Karnac, but I must be doing something wrong because after running a quick experiment, I always get very near the theorical value of 11.11% for each number from 1 to 9. Here is the procedure I followed.

A – I generated 55,000 random numbers between 10,000 and 99,999 using Excel RAND.BETWEEN function. The distribution of those numbers is fairly uniform.

B – Using Excel’s Data Analysis ToolPak, I picked out 1,000 of these numbers using the random sampling method.

C – From those 1,000 numbers, I isolated the leading number. Obviously this will give numbers between 1 and 9.

D – I then calculated the frequency of each number 1 to 9.

E – This was repeated 10 times and the average calculated. Here are the results I get:

For #1, average = 11.27% with a range from 09.00% to 13.00%
For #2, average = 11.32% with a range from 10.00% to 13.50%
For #3, average = 10.97% with a range from 09.30% to 12.50%
For #4, average = 10.88% with a range from 09.50% to 13.20%
For #5, average = 10.73% with a range from 09.50% to 11.60%
For #6, average = 10.74% with a range from 08.60% to 12.10%
For #7, average = 11.39% with a range from 09.00% to 13.30%
For #8, average = 11.32% with a range from 09.00% to 13.10%
For #9, average = 11.38% with a range from 09.10% to 13.90%

All values are around 11.11% and none approach 30.10% or the 4.58% values.

Now the question: WHAT I AM DOING WRONG?

Well, I think I must now read those web sites.

 

[tomtom]

I believe the theory is about general real occurrence of numbers…so since everything starts from 1 , it’s evidently that number 1 is more frequent. All elevators must have # 1, but many don’t have #20 for example, etc ….

I don’t think it can be useful in lottery at all…

 

[thornc]

Another good read!

They explain that this law only makes sense in "dimensioned" values! (1 inch, for example)!

Good read nevertheless!

 

[GillesD]

...
Hard to believe but true. We (some friends and I) have tested it on a sample size of 8,000, 55,000 and 65,000 numbers, and it all closely matches the above.
...

Karnac, could you provide some information on the nature of the samples you used to duplicate results for Benford's Law.

This could be the type of data, the distribution of data (uniform, normal or other), etc.

 

[Karnac]

Karnac, could you provide some information on the nature of the samples you used to duplicate results for Benford's Law.

This could be the type of data, the distribution of data (uniform, normal or other), etc.

GillesD....I only wish I could give you more , but as I said in the original post I found the article on the alt.rec.gambling newsgroup and pasted it here for the rest of the board to study to see if there were applications to the lotteries....sorry.

GillesD...here is the thread....LT can delete if he wishes or I will after you see it http://groups-beta.google.com/group/rec.gambling.lottery/browse_thread/thread/a94230d3a806135e

 

[Dennis Bassboss]

Benford's Law with low numbers... but Positionning has even more staggering data for low numbers...

 

[GillesD]

Thanks, Karnac, I misunderstood your initial posting on Benford’s Law and I was thinking it was you that ran an experiment on some samples. So I asked you about the data since I could not duplicate the results obtained.

I read the information provided on the site you indicated and I can see it was not you.

It seems that Benford’s Law apply for the first digit of a 6-number combination. So we have:
- a 1 if the first number is 1 or any number between 10 to 19;
- a 2 if the first number is 1 or any number between 20 to 29;
- a 3 if the first number is 1 or any number between 30 to 39;
- a 4 if the first number is 1 or any number between 40 to 49;
- a 5 if the first number is 5;
- a 6 if the first number is 6;
- a 7 if the first number is 7;
- a 8 if the first number is 8;
- a 9 if the first number is 9.

I decided to check how well Benford’s Law applies to the 2179 drawings of Lotto 6/49 and also to all possible 13,983,816 combinations. Here are the results I get:
# 1: 30.10% -- -- 36.67% -- -- 35.45%
# 2: 17.61% -- -- 14.69% -- -- 14.94%
# 3: 12.49% -- -- 09.45% -- -- 10.08%
# 4: 09.69% -- -- 09.32% -- -- 08.74%
# 5: 07.92% -- -- 07.34% -- -- 07.77%
# 6: 06.69% -- -- 06.10% -- -- 06.88%
# 7: 05.80% -- -- 06.70% -- -- 06.08%
# 8: 05.12% -- -- 04.82% -- -- 05.36%
# 9: 04.58% -- -- 04.91% -- -- 04.71%

The percentage given for each number (1 to 9) are respectively:
- first, the percentage given by Benford’s Law;
- then, the percentage obtained for the 2179 drawings of Lotto 6/49;
- and finally, the percentage obtained for the all possible combinations in a 6/49 lottery;

So, looking at it this way, it appears that Benford’s Law may apply somewhat for a 6/49 lottery but when considering all possible combinations, there are to be some substantial differences.

 

[Karnac]

Thanks GillesD for clearing that up...I knew I had something, but I didn't know what I had. Interesting numbers to say the least.