## eigenschmeigen 3 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?

1. satellite73

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

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

if n=2 P=1 if n>3 P<1 ....

4. UnkleRhaukus

$P(n)$

5. eigenschmeigen

ill post my thoughts so far.

6. eigenschmeigen

trivially P(1) = 1

7. UnkleRhaukus

this question reminds me of Buffon's needle problem

8. eigenschmeigen

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

now for three points

10. UnkleRhaukus

is P(3)=1/2?

11. UnkleRhaukus

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

yes i think so! maybe.. |dw:1333724818545:dw|

13. eigenschmeigen

placing C on the circle somewhere.. where is allowed?

14. eigenschmeigen

maybe its not 1/2 ...

15. UnkleRhaukus

1/2 in the worst case scenario, ie if the first two points are opposite

16. eigenschmeigen

|dw:1333725166177:dw| i think its $\frac{l}{(circumference)}$

17. eigenschmeigen

those lines from A and B are diameters

18. UnkleRhaukus

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

thanks for the help

20. rym

50 percent

21. TuringTest

I agree, good question This is perhaps a good question for the meta-math section

22. eigenschmeigen

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

whats the meta math section?

24. TuringTest

click the "mathematics" blue bar you will see it is a category

25. TuringTest
26. TuringTest

hard and irregular questions are found and solved there

27. eigenschmeigen

oh cool

28. TuringTest

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

or both? hehe

30. TuringTest

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

I'm not sure if you can post in both sections with the new "bump" system, but feel free to try :)

32. TuringTest

oh, mr.math is here after all maybe he has some nice thoughts on this

33. Mr.Math

I found this http://mathproblems.info/images/prob1.pdf

34. FoolForMath

35. TuringTest

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

I can see how they're kinda the same...

37. eigenschmeigen

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

gonna close it up now unless anyone has more to add, thanks guys for all your help :)