Might as well start off with the table produced in the part iii solution, in simplified form:
x in hex x^3 and 3^x leading digit decimal length
.1 to .1428... 1 .0162450656184296
.285... .32cb 1 no match with x
.6597f... .8 1 no match with x
2.148... 2.188d... 9 no match with x
...
3.a25d... 3.c91b... 3 0.151337335764465
...
Ellipses indicate none of the ranges' x values matched the functions' leading digits.
The two stretches on the number line total .1675824013828946.
That covers the numbers above .1 hex.
Below .1 hex, all 3^x begin with a 1, we need to know what fraction of that 1/16 length of number line has a leading hex digit 1 and has x^3 also begin with hex digit 1. This fraction, I'm sure is cuberoot(2)/16, so the portion of the number line (total length) is cuberoot(2)/256 ~= .004921566601151848.
This brings the total length of number line occupied to .1725039679840464.
As we're seeking the probability, this is divided by 16 to give .0107814979990029.

Posted by Charlie
on 20160620 10:37:02 