What is the Probability of Real Roots in a Quadratic Equation?

[matiasmorant]

I want to know how to compute the probability that a random quadratic equation of the form a x²+b x+c=0 has real roots. So, i want to find the probability that b²>4ac (assuming each variable has a continuous constant distribution in an interval (-R,R))

I know that the probability is about 0.627...



the general problem: what is the probability that a polinomial of degree n has x real roots?

 

[dacruick]

shouldn't the probability depend on R? And to clarify, your a, b, and c values are between -R and R right?

 

[matiasmorant]

yes, a, b, and c are values between -R and R. The probability COULD depend on R, but if you think about it, you will see it doesn't. make the calculation and R will cancel out.

to find the desired probability, the next step would be taking the limit as R approaches infinity, but since the probability doesn't depend on R, the probability for all quadratic equations between (-infinity, infinity) is the same as the probability for the equations between (-R , R)

 

[matiasmorant]

now I know that the probability is 1/72 (41 + Ln[64]). numerical value0.627206709491106553562547121233...

but I'm not satisfied with the method I used. this is how I did it:
we want to find the probability that 0<b^2-4ac, all having uniform distribution in the interval (-R,R)
So now let's define a variable z=-4ac, such a variable would have a distribution $$\frac{\text{Ln}\left[16 R^2\right]-\text{Ln}[4|x|]}{8 R^2}\left(\mu \left[4 R^2+x\right]-\mu \left[x-4 R^2\right]\right)$$

and the variable w=b^2 has a distribution$$\frac{\mu [x]-\mu \left[x-R^2\right]}{2 R \sqrt{x}}$$

were $$\mu$$ is the unit step function

now the probability distribution of the discriminant D=b^2-4ac=z+w is the convolution of the probability distributions of z and w, which yields a very large result.

now (to compute the probability that D>0 )we have to integrate the probability distribution of D from 0 to infinity; which yields the previous result. I could only do it using Mathematica

 

[disregardthat]

Since the set of real numbers has infinite measure, I don't think you can have a uniform distribution on them.