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        riddles >> medium >> The probability of distance between roots
        
(Message started by: Michael_Dagg on Sep 17^th, 2007, 3:41pm)

Title: The probability of distance between roots
Post by Michael_Dagg on Sep 17^th, 2007, 3:41pm

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?

Title: Re: The probability of distance between roots
Post by Barukh on Sep 18^th, 2007, 12:53am

[hide]1/3[/hide]?

Title: Re: The probability of distance between roots
Post by Michael_Dagg on Sep 18^th, 2007, 8:44am

Correct! Did you find it by of computing the area of the intersection
of a region between two parabolas and a square?

Title: Re: The probability of distance between roots
Post by Barukh on Sep 18^th, 2007, 10:50am

on 09/18/07 at 08:44:02, 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.

:-/

Title: Re: The probability of distance between roots
Post by Michael_Dagg on Sep 18^th, 2007, 12:10pm

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 .

Title: Re: The probability of distance between roots
Post by JP05 on Sep 18^th, 2007, 7:05pm

This is interesting. I have to guess that you get the parabolas from the inequality by putting y=c and x=b?

Title: Re: The probability of distance between roots
Post by Barukh on Sep 19^th, 2007, 2:22am

on 09/18/07 at 12:10:47, 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 09/18/07 at 19:05:54, 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.