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(2): answers by Dej Mar)

"You mean negligible, and may as well be zero."

I mean: Integral{10 to 10} (any function) dx = 0, as the antiderivative at 10 is subtracted from itself, and even in a discontinuous function with no proper antiderivative, the width of the Riemann sum element has gone to the limiting case of 0.

Of course in limited precision simulations, there would be a finite chance of zero or 10 as only a finite number of decimal or binary places are randomized. But in theory the probability is indeed zero.

Posted by Charlie on 2011-03-30 14:20:40