For a certain interval on the Real Axis, i.g.
[0, 1) if you randomly (with uniform distribution) pick a dot on it , what are the chances that number has in the decimal representation a finite number of digits on the fractional part?
My intuition says that it doesn't matter what interval you pick and the problem is the same as finding the ratio between numbers with a finite digits and numbers with infinite digits (either periodic decimal expansion or irrational). Is this correct?
Bonus: Is the ratio the same for all bases (e.g. base 2, base 16)?
This is just something I was wondering about.
My train of thought: You pick a dot on the real axis. You figure the first decimal by splitting the axis into segments of
0.1 (mark every
0.1). You have the segment your number is on. For example
[0.4, 0.5). I think it's a probabilistic impossibility to pick a dot exactly on the segment end, thus having exactly one fractional digit.
My rational is this:
- probability of being on a segment is the same for all segments.
- on the segment it is: Probability your dot is exactly on he segment end:
- on the segment it is: Probability your dot is not on the segment end: infinite.
So your dot wold be between two marks. In our example your number is of form
To figure out the second digit you split the interval further into ten (segments of length
0.01). For example your number is now on the segment
[0.48 - 0.49) By the same rationale it's a probabilistic impossibility that your dot is exactly on
0.48. So your number is
And you repeat the steps at infinit, figuring out more digits infinitely.
But (by this raionale) the final probability of your dot being a number with finite digits is actually
0 * ∞ (I think). So this is where I get stack overflow. More seriously it reminded me of some of the Zeno's paradoxes. Maybe this is a variant of one of them.
I am really curious about the answer to this.