For small numbers of a given length of digits, you can count the ones that contain the digit '3'. You can see these values in the following table:
| Number of digits | Examples | % Containing 3 |
|---|---|---|
| 1 (0-9) | 3 | 10% (1) |
| 2 (0-99) | 3,13,23,30,31,32,33,34,35,36,37,38,39,43,53,... | 19% (19) |
| 3 (0-999) | 3,13,...,30-39,...,300-399,430-439,... | 27% (271) |
You can count them by hand, but the count 'C' for any given number of digits 'n' is given by the following formula:
C = 10^n - 9^n
Thus the percentage is given by:
% = (10^n - 9^n) / 10^n
% = (10^n / 10^n) - (9^n / 10^n)
% = 1 - (9/10)^n
Taking the limit of this a n -> inf and you see that % = 1.
James Grimes does an excellent job of explaining it in this video:
https://www.youtube.com/watch?v=UfEiJJGv4CE
