A random quadratic with real roots, Geometric Probability

Problem 4.1

Let $p, q$ be independent uniform random variables in $[-1, 1]$ . What is the probability that the quadratic $x^2 + 2 p x + q = 0$ has real roots?

Solution. The quadratic has real roots iff its discriminant is non-negative, i.e., $(2p)^2 - 4q \geq 0$ , or equivalently $q \leq p^2$ . The parameter space is the square $[-1,1]^2$ , of area $4$ .

The region $\{(p,q) : q \leq p^2\}$ inside the square is everything below the parabola $q = p^2$ . Its area is $\int_{-1}^{1} \big( p^2 - (-1) \big) \, dp = \int_{-1}^{1} (p^2 + 1) \, dp = \tfrac23 + 2 = \tfrac83.$ Hence the probability is $\boxed{\mathbb{P} = \frac{8/3}{4} = \tfrac23.} \qedhere$

Random quadratic x^2 + 2px + q with (p,q) uniform in [-1,1]^2 . The quadratic has real roots iff (p,q) lies below the parabola q = p^2 (sage region, area \tfrac83 ). The complementary region (copper) has area \tfrac43 . The ratio gives \mathbb{P} = \tfrac23 . — Random quadratic $x^2 + 2px + q$ with $(p,q)$ uniform in $[-1,1]^2$ . The quadratic has real roots iff $(p,q)$ lies below the parabola $q = p^2$ (sage region, area $\tfrac83$ ). The complementary region (copper) has area $\tfrac43$ . The ratio gives $\mathbb{P} = \tfrac23$ .

import numpy as np
p, q = np.random.uniform(-1, 1, (2, 10**7))
real_roots = p**2 >= q
print(f"sim: {real_roots.mean():.5f}   exact: {2/3:.5f}")
# sim: 0.66659   exact: 0.66667