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.
The easiest one first:
x and x^2
Scaling (that is, order of magnitude) has no effect on x^2, so .1 to .999... work the same percentages as 1 to 9.999... and .01 to .09999.... So let's work with 1 through 9.9999... (equal to 10).
A 1 matches a 1 from 1 to sqrt(2).
Any number beginning with 2 through 7 will not have a match.
An 8 matches an 8 from sqrt(80) to 9.
A 9 matches a 9 from sqrt(90) to 10.
The total probability of a match from 1 to 10 is therefore:
(sqrt(2)  1 + 9  sqrt(80) + 10  sqrt(90)) / 9
= (18 + sqrt(2)  sqrt(80)  sqrt(90)) / 9
~= .109234296874307
As mentioned, the probability is the same if x were between .1 and .9999..., etc. So regardless of the relative probabilities of falling into one of these orders of magnitude, the probability is the same as the conditional probability: (18 + sqrt(2)  sqrt(80)  sqrt(90)) / 9 ~= .109234296874307.

Posted by Charlie
on 20110329 17:03:54 