For a randomly chosen real number x on the interval (0,10) find the exact probability of each:
(1) That x and 2^{x} have the same first digit
(2) That x and x^{2} have the same first digit
(3) That x^{2} and 2^{x} have the same first digit.
(4) That x, x^{2} and 2^{x} all have the same first digit.
First digit refers to the first nonzero digit of the number written in decimal form.
(In reply to
An attempt as (3), but it looks wrong according to simulation by Charlie)
You wrote:
Out of the range .1 to .99..., those x's from .1 to sqrt(2)/10 have
squares beginning with a 1. Thus the probability within this range is
(sqrt(2)/10  .1) / .9. But this same probability also applies in each
of the infinitely many orders of magnitude for x, and so (sqrt(2)/10 
.1) / .9 is the correct probability for all of the 0 to .999999...
range.
You missed some. Numbers on the interval (sqrt(.1),sqrt(.2)) also have square beginning with a 1. Im not sure how this affects your result. I dont have time to look into it further right now.

Posted by Jer
on 20110330 14:10:39 