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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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 re(3): answers Comment 12 of 12 |

"You mean negligible, and may as well be zero."

I mean: Integral{10 to 10} (any function) dx = 0, as the antiderivative at 10 is subtracted from itself, and even in a discontinuous function with no proper antiderivative, the width of the Riemann sum element has gone to the limiting case of 0.

Of course in limited precision simulations, there would be a finite chance of zero or 10 as only a finite number of decimal or binary places are randomized. But in theory the probability is indeed zero.

 Posted by Charlie on 2011-03-30 14:20:40

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