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eigenschmeigen

  • 2 years ago

n points are placed at random on the circumference of a circle, what is the probability that they all lie within a common semicircle?

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  1. satellite73
    • 2 years ago
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    i have googled this question and there are many explanations, but i have to say i don't even really understand what it is asking

  2. eigenschmeigen
    • 2 years ago
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    think of placing points at random, now what is the probability that it is possible to cut the circle in half leaving all the points on one side, for example it is possible here: |dw:1333717099690:dw|but not here:|dw:1333717241684:dw|

  3. UnkleRhaukus
    • 2 years ago
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    if n=2 P=1 if n>3 P<1 ....

  4. UnkleRhaukus
    • 2 years ago
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    \[P(n)\]

  5. eigenschmeigen
    • 2 years ago
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    ill post my thoughts so far.

  6. eigenschmeigen
    • 2 years ago
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    trivially P(1) = 1

  7. UnkleRhaukus
    • 2 years ago
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    this question reminds me of Buffon's needle problem

  8. eigenschmeigen
    • 2 years ago
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    P(2) is also easy, but we can learn from it: 1 point, A is already there, so placing another point the only place which is questionable is 180 degrees from A , i dont know whether it counts but it doesnt matter as the probability of that position exactly is 0 , therefore P(2) =1

  9. eigenschmeigen
    • 2 years ago
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    now for three points

  10. UnkleRhaukus
    • 2 years ago
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    is P(3)=1/2?

  11. UnkleRhaukus
    • 2 years ago
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    the first two points are in the same semi circle always , the third is either in-between the or out-between them, equal chance/?

  12. eigenschmeigen
    • 2 years ago
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    yes i think so! maybe.. |dw:1333724818545:dw|

  13. eigenschmeigen
    • 2 years ago
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    placing C on the circle somewhere.. where is allowed?

  14. eigenschmeigen
    • 2 years ago
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    maybe its not 1/2 ...

  15. UnkleRhaukus
    • 2 years ago
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    1/2 in the worst case scenario, ie if the first two points are opposite

  16. eigenschmeigen
    • 2 years ago
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    |dw:1333725166177:dw| i think its \[\frac{l}{(circumference)}\]

  17. eigenschmeigen
    • 2 years ago
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    those lines from A and B are diameters

  18. UnkleRhaukus
    • 2 years ago
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    this is a good question . and i can see you are getting closer and closer, but for now i must go (you might want to check out the Buffon's needle problem for some hints

  19. eigenschmeigen
    • 2 years ago
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    thanks for the help

  20. rym
    • 2 years ago
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    50 percent

  21. TuringTest
    • 2 years ago
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    I agree, good question This is perhaps a good question for the meta-math section

  22. eigenschmeigen
    • 2 years ago
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    to anyone viewing i think (guessing) maybe i should find l in terms of theta , then use an integral to find P(theta from A diameter)

  23. eigenschmeigen
    • 2 years ago
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    whats the meta math section?

  24. TuringTest
    • 2 years ago
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    click the "mathematics" blue bar you will see it is a category

  25. TuringTest
    • 2 years ago
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    http://openstudy.com/study#/groups/Meta-math

  26. TuringTest
    • 2 years ago
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    hard and irregular questions are found and solved there

  27. eigenschmeigen
    • 2 years ago
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    oh cool

  28. TuringTest
    • 2 years ago
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    but it can take a long time to get a response in that section, so take your pick keep "bumping" it here, or post in meta-math and wait...

  29. eigenschmeigen
    • 2 years ago
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    or both? hehe

  30. TuringTest
    • 2 years ago
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    in the meantime let me call on some who may be able to solve this: @across @JamesJ @Zarkon @Mr.Math @FoolForMath interesting probability problem (none are online right now it seems)

  31. TuringTest
    • 2 years ago
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    I'm not sure if you can post in both sections with the new "bump" system, but feel free to try :)

  32. TuringTest
    • 2 years ago
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    oh, mr.math is here after all maybe he has some nice thoughts on this

  33. Mr.Math
    • 2 years ago
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    I found this http://mathproblems.info/images/prob1.pdf

  34. FoolForMath
    • 2 years ago
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    I liked this answer: http://math.stackexchange.com/a/18371/2109

  35. TuringTest
    • 2 years ago
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    I can almost understand the MSE one, but I can't read Mr.Math's answer... when will my brain grow up like that?

  36. TuringTest
    • 2 years ago
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    I can see how they're kinda the same...

  37. eigenschmeigen
    • 2 years ago
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    i understand FFMs but i am struggling with mr.maths , i understand what's going on in general, im just not following all of the maths there. i think i need more experience with continuous probability and expected values..

  38. eigenschmeigen
    • 2 years ago
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    gonna close it up now unless anyone has more to add, thanks guys for all your help :)

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