/sci/ - Science & Math

Static fire ~Feb. 12th, launch ~Mar. 10th?
I would like to gently remind everyone that March 10th is the 69th day of the year.

>>12509794
P = (size of subset of digits / numeric base)^(digits) = (S/B)^D only works if you specify a particular subset, like {5,6,9}.

if you want to answer "what is the chance that a bill contains any subset of only three digits?", then you got to multiply the previous formula by the number of distinct 3 digit subsets you can form from 10 digits without duplicates (i.e {1, 2, 3} is ok, but {1, 1, 1} is not). this is 10 choose 3 = (10 3). so P(D digit number is formed from exactly 3 unique digits) = (B S) * (S / B) ^ D.

note, this ignores digits like 11111111. if you want to include these, then you have to sum the above formula for all smaller subsets. thus, the final formula will be [eqn]P = \sum_{i=1}^S \begin{pmatrix}B\\ i\end{pmatrix} \left(\frac{i}{B}\right)^D[/eqn]

so to answer your question...
the number of 8=D digit strings composed of a subset of exactly 3 decimal digits is P = (10 3) * (3 / 10)^8 ~= 0.0078732
and the number of 8=D digits strings composed of 3 or fewer decimal digits is
P = (10 3) * (3 / 10)^8 + (10 2) * (2 / 10)^8 + (10 1) * (1 / 10)^8 ~= 0.0079885 (i tiny bit larger)