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 part (1) not so hard after all (spoiler) by Charlie)

Since the first digit cannot be zero,  it is actually 1/9 of the numbers on the interval (0,1) that begin with the digit 1, not 1/10.

Your solution is too small by 1/900.

Posted by Jer on 2011-03-30 10:01:43