Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

hartnn Group Title

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.

  • one year ago
  • one year ago

  • This Question is Closed
  1. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1351969912033:dw| Something like this.

    • one year ago
  2. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    yes, thats the case when centre does not line inside circle...

    • one year ago
  3. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    *lie

    • one year ago
  4. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    and i need probability

    • one year ago
  5. AccessDenied Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I wonder if coordinate geometry could be helpful here? To get a name on the center point and the two random points...

    • one year ago
  6. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    i tried taking centre as origin and using distance formula....but its getting too complicated and no hopes of finding probability

    • one year ago
  7. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    *centre of square

    • one year ago
  8. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    What is the condition that the point is in the circle? In the coordinates? If you found it, please write.

    • one year ago
  9. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw: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 ?

    • one year ago
  10. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    It is a very good condition! Try to use it in polar system.

    • one year ago
  11. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Two points with coordinates \((\rho_1,\varphi_1)\) and \((\rho_2,\varphi_2)\).

    • one year ago
  12. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    my brain suddenly goes into hibernate mode when it comes to polar co-ordinates :P and remember we have to find probability.

    • one year ago
  13. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1351970854925:dw|

    • one year ago
  14. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    If 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?

    • one year ago
  15. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    that 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...

    • one year ago
  16. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Now 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:dw|Any 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.

    • one year ago
  17. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    u lost me here....

    • one year ago
  18. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    how square of side 2pi ?

    • one year ago
  19. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    We 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..

    • one year ago
  20. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Now 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?

    • one year ago
  21. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    You have only to plot this condition in the square. Please, say something.

    • one year ago
  22. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1351972589720:dw|

    • one year ago
  23. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    No, 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|

    • one year ago
  24. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    filled area = 9pi^2/ 8 ?

    • one year ago
  25. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    filled area = \((2\pi-\frac{\pi}2)^2\)

    • one year ago
  26. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    oh, yes....multipled by 2, twice, instead of once....so 9pi^2/16

    • one year ago
  27. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    sorry, 9pi^2/4

    • one year ago
  28. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Now it is OK.

    • one year ago
  29. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Now divide it by the area of the whole square and get final probability.

    • one year ago
  30. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    thats 9/16...but i am trying to understand that figure.....

    • one year ago
  31. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    i still don't understand why is the side of square =2pi

    • one year ago
  32. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Can 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|

    • one year ago
  33. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    randomly with uniform distribution ?

    • one year ago
  34. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    and no.

    • one year ago
  35. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    It 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.

    • one year ago
  36. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    So what with sniper?

    • one year ago
  37. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    2 quares of side 2R and R ?

    • one year ago
  38. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    Yes. They are concentric circles with radiuses R and 2R.

    • one year ago
  39. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    @CalebBeavers @AccessDenied @geerky42 Anyone got it?

    • one year ago
  40. geerky42 Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    1/2 right?

    • one year ago
  41. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    @geerky42 No.

    • one year ago
  42. hartnn Group Title
    Best Response
    You've already chosen the best response.
    Medals 3

    how does a circle become a square? ...... nevermind, got it.....i will go through this again later, i can't keeps my eyes open now.

    • one year ago
  43. AccessDenied Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Prob 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.. :)

    • one year ago
  44. klimenkov Group Title
    Best Response
    You've already chosen the best response.
    Medals 2

    @AccessDenied That's right. Did you get the very first question? About 2 points?

    • one year ago
  45. AccessDenied Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I was not able to get it. I was trying my own method but couldn't figure it out. :(

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.