Geometry Puzzle: Center of Square in Circle(pratikpoddarcse.blogspot.in)
pratikpoddarcse.blogspot.in
Geometry Puzzle: Center of Square in Circle
http://pratikpoddarcse.blogspot.in/2012/11/geometry-puzzle-center-of-square-in.html
3 comments
Quora has a great discussion on this: http://www.quora.com/Mathematics/What-is-the-probability-of-...
Thanks for the link, but Quora won't let me read it unless I sign in, and I just don't feel like doing that on my iPad. Requiring a log in to read the page seems a bit backwards thinking to me.
Ahh, looks like this link explains why Quora has started sucking: http://www.businessinsider.com/the-sudden-mysterious-exit-of...
Not a surprising story. So many startups pursue money and exits over building a great product. Pushing out people who made the product awesome to begin with. Now Quora can be counted among them. Sad really.
thanks for the link vecter.
The problem is ill-posed: What is the probability of choosing does not describe the choosing process. For example: It turns out that I always choose my 2 points so that the center-of-the-square is never inside the circle, so the probability is exactly 0.
Not the answer that anyone is looking for, but shows the importance of understanding what is asked. Edit: and learning to specify exactly what you mean when you ask a question. As the saying goes: If you do not learn to write what you mean, then you will have trouble meaning what you write.
Not the answer that anyone is looking for, but shows the importance of understanding what is asked. Edit: and learning to specify exactly what you mean when you ask a question. As the saying goes: If you do not learn to write what you mean, then you will have trouble meaning what you write.
I completely agree with this and always find it instructive to examine wording thoroughly when posing problems or solutions. However, I think you could come across more likably by offering a better wording yourself in your comment. I'll have a shot at it:
"Given two random points inside a square, what is the probability that the center of the square lies in the circle formed by taking the points as diameter?"
I'm not happy with my opening clause there though, possibly because I omitted to explicitly state the distribution (that is, uniform - all points are equally likely). It seems there is a convention to assume uniform when no other information is given, but I still don't like "random" there. What would be your wording?
"Given two random points inside a square, what is the probability that the center of the square lies in the circle formed by taking the points as diameter?"
I'm not happy with my opening clause there though, possibly because I omitted to explicitly state the distribution (that is, uniform - all points are equally likely). It seems there is a convention to assume uniform when no other information is given, but I still don't like "random" there. What would be your wording?
Changed the problem statement. Thanks
No, that is not the answer that anyone is looking for.
The points i.i.d. with respect to Lebesgue measure on the square. Happy?
Edited. Thanks a ton.
if you look at the possibilities you can take one approach of dividing the square into four quadrants (like a window) and from that you can see that any point located in a diagnola opposing will make a hit on the circle. So from that we know that to pick a spot in one of the four quadrants is 1 in 4 and to pick the disagonal would be another 1 in 4 chance. As the first picks location is only relevant for the second pick then it is really just working out the odd's for the sencond being diagonal.
But there's more. It would be possible for two locations on the same level dependant upon there location to also make a circle that encompased the centre fo the square. So what about those permutations, well is you divide those squares again into four and look at it then you start thinking, this could get into some recuring details and have an urge to push square pegs into round holes phsyicaly as well as mentaly :).
That all said anything with a circle has to involve PI, even if you end up eating a entire pie just to work it out. So as the square has four sides I'm going to say the answear is 4 multiplied by PI and accept I'm probably wrong but it is what we call a educated guess.
Though you can imagine after being asked that and finding the right answear to only be asked; "Now using 3d points, what are the chances of you forming a sphere that would encompass the centre of a cube", you just know it. Also would show how adaptable there solution is.