1 of 38

A Rainbow of Random Digits�

Ryan Pietropaolo

TCM 2024

2 of 38

Introductory Problem

Let’s create triangles!!

You are given two wooden dowels, one twice as long as the other. Suppose that you break the longer dowel in a random location to form two additional dowels. What is the probability the three dowels form a triangle when placed end-to-end?

​

3 of 38

Where could we cut the longer dowel so that the resulting 3 dowels do NOT form a triangle?

4 of 38

Geometric Probability (2D)

  • Suppose two numbers are chosen at random, where both are between 0 and 4.
  • a) What is the probability that their sum exceeds their product?
  • b) What is the probability that their sum is between 5 and 6?

​

5 of 38

Sample Space

6 of 38

 

​

​

​

7 of 38

b) 5 < x + y < 6

 

8 of 38

Now to our Problem

  • Given two random numbers between 0 and 1. What is the probability that the first non-zero digit of their ratio is a 1? 2? 3?...9?

​

  • First non-zero digit???
  • ex) .0036378 23.499 0.100045

​

9 of 38

Let’s simplify the problem…�Case 1 : y/x < 1

  •  

10 of 38

Which can be re-written as:

  •  

11 of 38

What are we interested in?

 

12 of 38

Calculus Approach

  •  

13 of 38

Geometric Series

  •  

14 of 38

What are the areas of the other colors where y/x < 1 ?

15 of 38

How about 6? Give it a try…

  •  

16 of 38

Thank you Copy and Paste!

  •  

17 of 38

The probability is 1/18 for all of the 9 leading digits!

  •  

18 of 38

Case 2: y/x > 1

  •  

19 of 38

How should we find the area of the yellow regions above y=x?

20 of 38

Triangles!!!

  •  

21 of 38

Generalize

  •  

22 of 38

Your turn…

  • Determine the probability that the first non-zero digit is a 6 when the ratio is greater than 1.

23 of 38

Generalize

  •  

24 of 38

Final Probability Distribution

  • 100 random ratios from Python.

​

25 of 38

Benford’s Law

  • Also known as the “First Digit Law”. When real-life sets of numerical data are observed, the leading digit is likely to be small. (Wikipedia)

26 of 38

Random Digit: 3-Space

  •  

27 of 38

Case 1: xy/z > 1

 

28 of 38

Finding Volume!

  •  

29 of 38

Geometric Series for each value k

  •  

30 of 38

Case 2: xy/z < 1

 

31 of 38

 

  •  

 

32 of 38

Similarly, the volume of interest under z = xy / .4

  •  

33 of 38

So, the volume or probability that the xy/z is between .3 and .4 is…

  •  

34 of 38

Now generalize…

  •  

35 of 38

 

  •  

36 of 38

Finally putting both cases together…

  •  

37 of 38

References

  • S.H. Freidberg, The Distribution of First Digits, The College Mathematics Journal, Volume 15, Number 2, (1984) 120-125
  • R.A. Raima, The First Digit Problem, American Mathematics Monthly, 83 (1976) 521-538
  • Dan Teague, A Random Digit Problem, NCSSM

​

38 of 38

Thanks for coming!�Enjoy the rest of the conference!

​

​

  • pietropaolo@ncssm.edu

​