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 re: answers by Charlie)

"Since there are indeed an infinite number of points, the probability of an end point being chosen, even in the closed interval case, is zero."

You mean negligible, and may as well be zero.

I see that my initial observation was incorrect in that I incorrectly included/excluded those matches that began with zero and not a 'first non-zero' atch.  My recalculation for (1) matched precisely with yours.  My estimates for (2), (3) and (4) are at a difference, but probably due to errors in my calculations for the 'first non-zero' matches.

Posted by Dej Mar on 2011-03-30 13:10:44