Most Random Number

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Most Random Number

• Tom Carter
• CSSS
• 2004

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Heres a somewhat strange question . . .

• What is the most random number between 1 and 1000?

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How to go at it?

• Lets try to develop a probability distribution
  function representing some general notion of a
  randomness property for numbers . . . -)

4
First guess everything is normal, right?
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• Too simple. We do get a unique maximum at 500,
  but that cant be right.
• As everybody knows, you never get the actual data
  value. There are always errors that push you
  away from the real value.
• So, we should overlay a repelling error
  dark-force on top.

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Error dark-force
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Overlay the dark-force on the normal distribution
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• Now were getting somewhere!
• But, we dont have a unique maximum any more.
• There must be a bias in one direction or the
  other, mustnt there?

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• Aha! We forgot about Benfords law on the
  distribution of significant digits.
• One form of Benfords law says that on average,
  the proportion of numbers having first digit less
  than d will be about log(d1) for d 1, 2, . .
  ., 9.
• Another way to say this is that the distribution
  will be approximately 1/(d1), so well overlay a
  power law

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1/(x/100 1)
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Normal, with dark-forceand digit overlay
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• Now were almost done. We have a single maximum,
  and it is not the simplistic 500.
• One more piece. Random numbers must not have
  special properties, right?
• So, they probably wont be divisible by small
  primes like 2 or 3 or 5.
• Lets write a couple of small MatLab functions
  for all this.

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function r mrn(x) "most random number"
function -) r (1/((x/100) 1)) (1 -
normpdf(x, 500, 70) normpdf(500,500,70))
normpdf(x, 500, 167)
function r maxrn search for "most random
number" using mrn(x) maxval 0 maxrn 0 for n
11000 if rem(n,2) 0 rem(n,3) 0
rem(n,5) 0 mrn(n) gt maxval
maxval mrn(n) maxrn
n end end r maxrn
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Now we run our function, and there it is, the
most random number between 1 and 1000
gtgt maxrn ans 347
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• What about more general versions of this?
• In other words, what about the most random number
  between 1 and 10, 1 and 100, 1 and 10000, etc.?

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function r mrng(x,maxn) "most random number"
function -) r (1/((10x/maxn) 1)) (1 -
normpdf(x, maxn/2, 0.07maxn)/normpdf(maxn/2,maxn/
2,0.07maxn)) normpdf(x, maxn/2, maxn/6)
function r maxrng(maxn) search for "most
random number" using mrng(x,maxn) maxval
0 maxrn 0 for n 1maxn if rem(n,2) 0
rem(n,3) 0 rem(n,5) 0 mrng(n,maxn) gt
maxval maxval
mrng(n,maxn) maxrn n
end end r maxrn
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Summary of most random numbers

• 1 to 10 7
• 1 to 50 17
• 1 to 100 37
• 1 to 500 173
• 1 to 1000 347
• 1 to 10,000 3,479
• 1 to 100,000 34,799
• 1 to 1,000,000 347,971