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.
(In reply to
answers by Dej Mar)
"As a real number line has an infinite number of points an exact probability can not be given for a randomly chosen real number but only an approximation. "
The events in question represent a finite length on the number line, representing finite total lengths on the real number line, and so an exact solution is possible. (though I mistrust my answer for part 3 as it disagrees with a simulation I did.)
"As we are given the notation for the interval as (0,10) and not [0,10], both endpoints, 0 and 10, are excluded."
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.

Posted by Charlie
on 20110330 09:56:27 