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
re: part (1) not so hard after all (spoiler) by Jer)
Thanks for the correction. I've amended the original comment with the correction and appropriate attribution to your finding it.
I'm still at a loss for the discrepancy between the analytic and simulation solutions for part 3.

Posted by Charlie
on 20110330 10:57:13 