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 x² have the same first digit

(3) That x² and 2^x have the same first digit.

(4) That x, x² and 2^x all have the same first digit.

First digit refers to the first non-zero 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 2011-03-30 14:10:39