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wu :: forums    riddles    medium (Moderators: Grimbal, SMQ, ThudnBlunder, william wu, towr, Icarus, Eigenray)    The probability of distance between roots « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: The probability of distance between roots  (Read 708 times) Michael Dagg Senior Riddler     Gender: Posts: 500 The probability of distance between roots   « on: Sep 17th, 2007, 3:41pm » Quote Modify Let  b,c  be real numbers randomly chosen in [0,1].   What is the probability that the distance in the complex   plane between the two roots of the eqaution   z2  + bz  +  c = 0  is not greater than 1? IP Logged Regards, Michael Dagg Barukh Uberpuzzler     Gender: Posts: 2276 Re: The probability of distance between roots   « Reply #1 on: Sep 18th, 2007, 12:53am » Quote Modify 1/3? IP Logged Michael Dagg Senior Riddler     Gender: Posts: 500 Re: The probability of distance between roots   « Reply #2 on: Sep 18th, 2007, 8:44am » Quote Modify Correct! Did you find it by of computing the area of the intersection   of a region between two parabolas and a square? IP Logged Regards, Michael Dagg Barukh Uberpuzzler     Gender: Posts: 2276 Re: The probability of distance between roots   « Reply #3 on: Sep 18th, 2007, 10:50am » Quote Modify on Sep 18th, 2007, 8:44am, Michael_Dagg wrote: Did you find it by of computing the area of the intersection   of a region between two parabolas and a square? Intersection? No, I found the area below a single parabola.     IP Logged Michael Dagg Senior Riddler     Gender: Posts: 500 Re: The probability of distance between roots   « Reply #4 on: Sep 18th, 2007, 12:10pm » Quote Modify y = (x2 + 1)/4  is the parabola you are referring to   over [0,1], and noting however that the distance between   the two roots is not greater than 1  iff  -1 < b2 - 4c < 1   which means that the point M(b,c) lies on the intersection   of the region in between the two parabolas   y = (x2 - 1)/4, y = (x2 + 1)/4 and the square x = 0,1  , y = 0,1 . « Last Edit: Sep 18th, 2007, 12:11pm by Michael Dagg » IP Logged Regards, Michael Dagg JP05 Guest Re: The probability of distance between roots   « Reply #5 on: Sep 18th, 2007, 7:05pm » Quote Modify Remove This is interesting. I have to guess that you get the parabolas from the inequality by putting y=c and x=b? IP Logged Barukh Uberpuzzler     Gender: Posts: 2276 Re: The probability of distance between roots   « Reply #6 on: Sep 19th, 2007, 2:22am » Quote Modify on Sep 18th, 2007, 12:10pm, Michael_Dagg wrote: y = (x2 + 1)/4  is the parabola you are referring to   over [0,1] Yes, that’s the way I approached it. I just noticed that when b2 – 4c > 0, then two real roots are always at the distance less than 1.   on Sep 18th, 2007, 7:05pm, JP05 wrote: This is interesting. I have to guess that you get the parabolas from the inequality by putting y=c and x=b? Precisely. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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