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Powerful Digit (Posted on 2011-03-29) Difficulty: 3 of 5
For a randomly chosen real number x on the interval (0,10) find the exact probability of each:

(1) That x and 2x have the same first digit

(2) That x and x2 have the same first digit

(3) That x2 and 2x have the same first digit.

(4) That x, x2 and 2x all have the same first digit.

First digit refers to the first non-zero digit of the number written in decimal form.

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: An attempt as (3), but it looks wrong according to simulation | Comment 11 of 12 |
(In reply to An attempt as (3), but it looks wrong according to simulation by Charlie)

You wrote:
Out of the range .1 to .99..., those x's from .1 to sqrt(2)/10 have squares beginning with a 1. Thus the probability within this range is (sqrt(2)/10 - .1) / .9. But this same probability also applies in each of the infinitely many orders of magnitude for x, and so (sqrt(2)/10 - .1) / .9 is the correct probability for all of the 0 to .999999... range.

You missed some.  Numbers on the interval (sqrt(.1),sqrt(.2)) also have square beginning with a 1.  Im not sure how this affects your result.  I dont have time to look into it further right now.

  Posted by Jer on 2011-03-30 14:10:39

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