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A lack of mathematical intuition is a serious problem for many people from engineering to biology to economics. It certainly plagued me throughout my engineerin

A lack of mathematical intuition is a serious problem for many people from engineering to biology to economics. It certainly plagued me throughout my engineering bachelors studies and is something I continually work to get better at. In my opinion, physics students learn the best framework for thinking and get a very good mathematical intuition. For example, here's a problem from an introductory QM book that really threw me for a loop when I was studying: A needle of length L is dropped at random onto a sheet of paper ruled with parallel lines a distance L apart. What is the probability that the needle will cross a line?

maged 11y ago
> A needle of length L is dropped at random onto a sheet of paper ruled with parallel lines a distance L apart. What is the probability that the needle will cross a line? Thickness of line is needed right? Otherwise P approaches 100% as thickness approaches 0?
- davmre 11y ago
  The lines are infinitely thin. Equivalently you can imagine the paper is divided into regions of width L, and the question is whether the needle will cross a region boundary (https://en.wikipedia.org/wiki/Buffon's_needle https://en.wikipedia.org/wiki/Buffon's_needle).
  - nonbel 11y ago
    I don't think that page explains it very well, but have poor math background. I imagined notebook paper with horizontal lines spaced L apart and then the needle dropping at any angle. When the needle is vertical the probability it cross a line is 1, when horizontal it is zero. The length of the needle L is the hypotenuse of a triangle. If we call the angle from horizontal x, the "height" of the needle can be anywhere within h=Lsin(x) for x between 0 and pi/2. The "lines" are like a sample of a point from a uniform distribution U with width L, and h is an interval inside U. The probability a number sampled from a distribution of width L will fall within interval h is h/L. Substituting for h gives p(cross|x) = sin(x). Then assuming the needle is equally likely to drop at any angle, for any one angle theta we get probability density p(theta=x) = 1/(pi/2-0)= 2/pi. The probability the needle drops at angle x AND crosses a line is the product of p(theta=x)p(cross|x)= (2/pi)sin(x). As mentioned, x can range between 0 and pi/2. To get the probability the needle drops at angle x1 OR x2 OR x3, etc and cross we need to sum all these. So take the integral of (2/pi)sin(x) from 0:pi/2. This gives 2/pi.
theworstshill 11y ago
99%?
ihartley 11y ago
https://en.wikipedia.org/wiki/Buffon%27s_needle https://en.wikipedia.org/wiki/Buffon%27s_needle for anyone that's interested.
openfuture 11y ago
Since the lines are parallel you can rephrase the problem: A circle of radius L is centered x far away from a border 0<x<L. This is because one end of the needle will always end in some zone (the center) and the other end will be L far away (the circle). How much of the 2pi boundary is outside the zone? When x -> 0 then it's going to be 50% since one boundary line becomes a tangent and the other goes through the middle. When we move x by k (e.g. f(x+k)) then 2k new points will be added on the left side while 2*k points leave the boundary on the right side. When x=L/2 then the boundary lines will split the circle in four equal parts (since they're tangent to the radius at r/2 on both sides) so intuitively its 50%.
sotojuan 11y ago
> A lack of mathematical intuition is a serious problem for many people from engineering to biology to economics. This is true. I can't really say why, but after my Discrete Mathematics class a lot of my Computer Science problems became a lot easier to reason about.
- onaclov2000 11y ago
  I will say that after my discrete math class we really only talked about how to write a proof, and well, it really didn't help for me. (I think we were supposed to get further into stuff, but well, the class wasn't paced well, new professor, etc).
graycat 11y ago
For such a problem, usually "at random" means a uniform distribution. But on the plane, there is no uniform distribution. So, the "paper" can't be the plane. So, it might be fair to ask the size of the paper and what happens with the needle near the edges? E.g., on a rectangular sheet of paper of finite size, the needle can land in a position so that it does not cross a line but would on a larger sheet of paper.

maged 11y ago

> A needle of length L is dropped at random onto a sheet of paper ruled with parallel lines a distance L apart. What is the probability that the needle will cross a line? Thickness of line is needed right? Otherwise P approaches 100% as thickness approaches 0?

davmre 11y ago
The lines are infinitely thin. Equivalently you can imagine the paper is divided into regions of width L, and the question is whether the needle will cross a region boundary (https://en.wikipedia.org/wiki/Buffon's_needle https://en.wikipedia.org/wiki/Buffon's_needle).
- nonbel 11y ago
  I don't think that page explains it very well, but have poor math background. I imagined notebook paper with horizontal lines spaced L apart and then the needle dropping at any angle. When the needle is vertical the probability it cross a line is 1, when horizontal it is zero. The length of the needle L is the hypotenuse of a triangle. If we call the angle from horizontal x, the "height" of the needle can be anywhere within h=Lsin(x) for x between 0 and pi/2. The "lines" are like a sample of a point from a uniform distribution U with width L, and h is an interval inside U. The probability a number sampled from a distribution of width L will fall within interval h is h/L. Substituting for h gives p(cross|x) = sin(x). Then assuming the needle is equally likely to drop at any angle, for any one angle theta we get probability density p(theta=x) = 1/(pi/2-0)= 2/pi. The probability the needle drops at angle x AND crosses a line is the product of p(theta=x)p(cross|x)= (2/pi)sin(x). As mentioned, x can range between 0 and pi/2. To get the probability the needle drops at angle x1 OR x2 OR x3, etc and cross we need to sum all these. So take the integral of (2/pi)sin(x) from 0:pi/2. This gives 2/pi.

theworstshill 11y ago

99%?

ihartley 11y ago

https://en.wikipedia.org/wiki/Buffon%27s_needle https://en.wikipedia.org/wiki/Buffon%27s_needle for anyone that's interested.

openfuture 11y ago

Since the lines are parallel you can rephrase the problem: A circle of radius L is centered x far away from a border 0<x<L. This is because one end of the needle will always end in some zone (the center) and the other end will be L far away (the circle). How much of the 2pi boundary is outside the zone? When x -> 0 then it's going to be 50% since one boundary line becomes a tangent and the other goes through the middle. When we move x by k (e.g. f(x+k)) then 2k new points will be added on the left side while 2*k points leave the boundary on the right side. When x=L/2 then the boundary lines will split the circle in four equal parts (since they're tangent to the radius at r/2 on both sides) so intuitively its 50%.

sotojuan 11y ago

> A lack of mathematical intuition is a serious problem for many people from engineering to biology to economics. This is true. I can't really say why, but after my Discrete Mathematics class a lot of my Computer Science problems became a lot easier to reason about.

onaclov2000 11y ago
I will say that after my discrete math class we really only talked about how to write a proof, and well, it really didn't help for me. (I think we were supposed to get further into stuff, but well, the class wasn't paced well, new professor, etc).

graycat 11y ago

For such a problem, usually "at random" means a uniform distribution. But on the plane, there is no uniform distribution. So, the "paper" can't be the plane. So, it might be fair to ask the size of the paper and what happens with the needle near the edges? E.g., on a rectangular sheet of paper of finite size, the needle can land in a position so that it does not cross a line but would on a larger sheet of paper.