You have an infinite plane filled with infinite parallel lines, all parallel to the same line, each line a unit distance apart... kind of like a giant sheet of paper with straight lines drawn across at regular intervals.
You drop a needle of unit length (ie, the same length as the distance between the parallel lines) onto the infinite plane.
What is the probability that it will cross one of the lines?
Specifications:
* by "crossing a line", I mean having points along the needle that fall on either side of the line. So a needle that rests exactly along a line is NOT considered to cross the line. Neither is a needle that is perfectly vertical with its bottom point on one line and its top point on another (but all other vertical needles WILL cross one of the lines obviously)
* all points and lines are geometric, ie, infinitely thin
Would you, at first glance, expect this problem to be related to the number π (pi)? It is
This problem can be solved with calculus for sure - but I'm asking a secondary question: can the problem be solved WITHOUT calculus?
Question
unreality
You have an infinite plane filled with infinite parallel lines, all parallel to the same line, each line a unit distance apart... kind of like a giant sheet of paper with straight lines drawn across at regular intervals.
You drop a needle of unit length (ie, the same length as the distance between the parallel lines) onto the infinite plane.
What is the probability that it will cross one of the lines?
Specifications:
* by "crossing a line", I mean having points along the needle that fall on either side of the line. So a needle that rests exactly along a line is NOT considered to cross the line. Neither is a needle that is perfectly vertical with its bottom point on one line and its top point on another (but all other vertical needles WILL cross one of the lines obviously)
* all points and lines are geometric, ie, infinitely thin
This problem can be solved with calculus for sure - but I'm asking a secondary question: can the problem be solved WITHOUT calculus?
Good luck
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