Let be chosen at random from the interval . What is the probability that ?

Here denotes the greatest integer that is less than or equal to .

This is multiple choice, but I don't think posting the possibilities are necessary.

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- April 6th 2006, 04:25 PM #1

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- April 6th 2006, 06:52 PM #2

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- April 7th 2006, 06:12 AM #3

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Let, us solve, for

We have,

Thus, all solutions for are,

Thus, we that, all solutions satisfies

Iff,

.

Note, cannot be zero or positive because it would violate the inequality .

Thus, all work.

Thus,

thus, length =.15

length=.015

length=.0015

and so on "ad infinitum" (I am so cool using latin phrases).

Thus, we have the total length of the solutions to be:

this is a regular infinite geometric series.

My point is that, we can "intuitively" think of probability as the length of the success (which are the solutions) divided by the total possibilities (which is the length of interval). Thus, the probability is .