Solution
Real Roots at Random
Show the problem again
The coefficients a, b, c are chosen independently and uniformly from [0, 1]. What is the probability that the quadratic ax² + bx + c = 0 has real roots? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is (5 + 6 ln 2)/36 ≈ 0.2544. We need b² ≥ 4ac, i.e. ac ≤ b²/4. For two independent uniforms, P(ac ≤ t) = t − t ln t for 0 < t ≤ 1 (the area under the hyperbola ac = t in the unit square). Since b²/4 ≤ 1/4, we can use this directly: P = ∫₀¹ [b²/4 − (b²/4) ln(b²/4)] db. The first term is 1/12. For the second, −(1/4)∫₀¹ b²(2 ln b − ln 4) db = −(1/4)(−2/9 − (ln 4)/3) = 1/18 + (ln 4)/12. Adding, P = 1/12 + 1/18 + (ln 2)/6 = (5 + 6 ln 2)/36.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.