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.

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 2011-03-29 17:03:54