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 Powerful Digit (Posted on 2011-03-29)
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)

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 part (2) -- the easy one (spoiler for this part) | Comment 1 of 12

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

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