Yerushalmi

07-28-2013, 07:09 AM

Okay, so imagine you have a line segment:

--------------

Now divide it into n portions of equal size.

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|1|2|3|....|n|

Now take an identical line segment and divide it into n+1 portions of equal size (each of which is therefore slightly smaller than the first set of portions).

--------------

|1|2|....|n+1|

(ignore the fact that the n+1 portion is bigger because I had to fit more characters in it - in reality it's the same size as all the others)

Label each portion with its number, as I've already done above.

What percentage of the total length of the line segment has the SAME label under both numbering schemes?

(An alternative way to think about the problem: When you ask a computer's random number generator for a number between 1 and x, it doesn't actually pick a random number between 1 and x, instead it picks a random decimal between 0 and 1, multiplies by x, adds 1, and rounds down. So really, asking the computer for a random number is asking it to pick a random place on a line segment and tell you how far along the line segment it is. Imagine therefore you asked the computer to pick a random number between 1 and x, and then you suddenly realize that you made a mistake and you should've picked a random number between 1 and x+1. What are the chances that you would've gotten the same number?)

I thought of this question, and discovered the answer, two nights ago, but since it was Shabbat I couldn't write out the algebra and figure out why that's the answer. I only worked out the "why" this morning.

Anyway, the answer will be forthcoming after a little while for you folks to stew over it. If, of course, anybody here thinks math is awesome besides me :p

--------------

Now divide it into n portions of equal size.

--------------

|1|2|3|....|n|

Now take an identical line segment and divide it into n+1 portions of equal size (each of which is therefore slightly smaller than the first set of portions).

--------------

|1|2|....|n+1|

(ignore the fact that the n+1 portion is bigger because I had to fit more characters in it - in reality it's the same size as all the others)

Label each portion with its number, as I've already done above.

What percentage of the total length of the line segment has the SAME label under both numbering schemes?

(An alternative way to think about the problem: When you ask a computer's random number generator for a number between 1 and x, it doesn't actually pick a random number between 1 and x, instead it picks a random decimal between 0 and 1, multiplies by x, adds 1, and rounds down. So really, asking the computer for a random number is asking it to pick a random place on a line segment and tell you how far along the line segment it is. Imagine therefore you asked the computer to pick a random number between 1 and x, and then you suddenly realize that you made a mistake and you should've picked a random number between 1 and x+1. What are the chances that you would've gotten the same number?)

I thought of this question, and discovered the answer, two nights ago, but since it was Shabbat I couldn't write out the algebra and figure out why that's the answer. I only worked out the "why" this morning.

Anyway, the answer will be forthcoming after a little while for you folks to stew over it. If, of course, anybody here thinks math is awesome besides me :p