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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)

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Some Thoughts re: answers | Comment 6 of 12 |
(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 2011-03-30 09:56:27
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