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What's the probability you get real roots for x^2 + ax + b = 0, when a, b are randomly chosen from [0, 1]?
Keith Raskin ???? ???? ????
![Keith Raskin ???? ???? ????]()
Keith Raskin ???? ???? ????
Math Teacher, Author: Math, Humor, Fiction, Essay; FOLLOW ONLY, 30K connections ??
Ran across this problem at Brilliant.org, I think.
Real roots will occur when a^2 is greater than or equal to 4b, according to the discriminant. So, b must be less than or equal to (a^2)/4. We can use calculus to find the area under this curve, all values/coordinate pairs of x and y where y < (x^2)/4 as x ranges from 0 to 1. The integral (or antiderivative evaluated at 1) of the curve is (1^3)/12 = 1/12. Compare the area of values to the area of the box where x and y can range freely from 0 to 1, and you get the probability, 1/12. Unless I'm mistaken.
Found related problems elsewhere, including quora.com