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 2 years ago
The two points are uniformly and independently marked inside a square . What is the probability that the center of the square is inside the circle formed by the two points as the diameter.
 2 years ago
The two points are uniformly and independently marked inside a square . What is the probability that the center of the square is inside the circle formed by the two points as the diameter.

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klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1351969912033:dw Something like this.

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3yes, thats the case when centre does not line inside circle...

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0I wonder if coordinate geometry could be helpful here? To get a name on the center point and the two random points...

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3i tried taking centre as origin and using distance formula....but its getting too complicated and no hopes of finding probability

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2What is the condition that the point is in the circle? In the coordinates? If you found it, please write.

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3dw:1351970387637:dw the centre , if it lies on the circle(or inside it) will form 90 degree angle (or more) as shown.... but this doen't help, i think.....or does it ?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2It is a very good condition! Try to use it in polar system.

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Two points with coordinates \((\rho_1,\varphi_1)\) and \((\rho_2,\varphi_2)\).

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3my brain suddenly goes into hibernate mode when it comes to polar coordinates :P and remember we have to find probability.

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1351970854925:dw

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2If you have 2 points with polar coordinates (ρ1,φ1) and (ρ2,φ2), your condition will reduce the number of variables. \(\varphi_1\varphi_2\ge\frac{\pi}2\). Do you get it?

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3that i got... will that help in finding the probability ? also we still haven't used the info. that points are 'uniformly' marked....that is their pdf is of uniform distribution...

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Now try to get. \(\varphi_1\) and \(\varphi_2\) varies in \([0,2\pi]\). So, we can draw this as a square with a length equal to 2 pi:dw:1351971559152:dwAny point of this square will represent 2 points. One question is "What about \(\rho_1,\rho_2\)?" Actually the answer is that the probability depends only on \(\varphi\)  the angle and not on the distance. So, you just have to plot φ1−φ2≥π/2 in the square on the picture. Please do it.

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3how square of side 2pi ?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2We got a condition that the origin lies in the circle, right? It is \(\varphi_1\varphi_2\ge\frac{\pi}2\). That's ok? May be my English should be better..

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Now we use that fact that the points are uniformly marked. Randomly. So for a point it can be any angle from \([0,2\pi]\). So, it will be more demonstrably if we will plot values of this two angles is this way. It will be easier later to find probability. The angles of any point represent the point of the square. So we can say that any point of the square gives two angles of our random points. Got it?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2You have only to plot this condition in the square. Please, say something.

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2No, it will be like this. Just find the filled area and divide it by \(4\pi^2\)  the area of the square.dw:1351973075885:dw

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3filled area = 9pi^2/ 8 ?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2filled area = \((2\pi\frac{\pi}2)^2\)

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3oh, yes....multipled by 2, twice, instead of once....so 9pi^2/16

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Now divide it by the area of the whole square and get final probability.

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3thats 9/16...but i am trying to understand that figure.....

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3i still don't understand why is the side of square =2pi

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Can you solve this problem? The sniper hits randomly at the big circle. Any of his shot is in this circle. What is the probability to hit the small circle?dw:1351973773130:dw

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3randomly with uniform distribution ?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2It is because the angle of the point in polar coordinates varies from \(0\) to \(2\pi\). So any of this values can be the angle of the point in polar coordinates.

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.32 quares of side 2R and R ?

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2Yes. They are concentric circles with radiuses R and 2R.

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2@CalebBeavers @AccessDenied @geerky42 Anyone got it?

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3how does a circle become a square? ...... nevermind, got it.....i will go through this again later, i can't keeps my eyes open now.

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0Prob of hitting smaller circle = Area of smaller circle / Area of bigger circle = (pi R^2)/(pi (2R)^2) = 1/4 That's how I'd approach it, anyways.. :)

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.2@AccessDenied That's right. Did you get the very first question? About 2 points?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0I was not able to get it. I was trying my own method but couldn't figure it out. :(
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