Probability of Real Roots in a Quadratic Equation with Uniform(-θ,θ) Coefficients

Subhomoy Haldar
BySubhomoy Haldar• • Updated:
research publications academic probability mathematics statistics

Background

I wrote this paper towards the end of my Integrated Master of Science (IMSc) in Mathematics and Computing at Birla Institute of Technology, Mesra. Working under the guidance of Dr. Soubhik Chakraborty, I came across an interesting problem that had seen some discussion in the early 20th century but had been largely forgotten since.

The question is deceptively simple:

If you pick random numbers for the coefficients of a quadratic equation, what’s the chance that it has real roots?

This turned out to be a wonderful exercise in probability theory. Beyond the mathematics itself, the goal was to show that there are still unexplored corners in well-trodden areas of mathematics waiting to be revisited.

The Problem in Plain English

Consider the quadratic equation $Ax^2 + Bx + C = 0$. We know from school that it has real roots when the discriminant $B^2 - 4AC$ is non-negative. It takes an interesting angle when we ask: what if $A$, $B$, and $C$ are random numbers drawn independently from the same uniform distribution?

We looked specifically at the symmetric case where each coefficient comes from $U(-\theta, \theta)$, meaning any value between $-\theta$ and $+\theta$ is equally likely.

The Key Finding

The probability of obtaining real roots is approximately 62.7%.

This result comes from analysis of the probability distribution of $B^2$ and the conditions under which $B^2 \geq 4AC$. The calculation simplifies considerably when $AC \leq 0$ (which happens half the time), because then the discriminant is automatically non-negative.

We verified this theoretical result experimentally through Monte Carlo simulation, running thousands of random trials to confirm the calculation.

Abstract

In this paper, we seek to find out the probability of obtaining real roots of a quadratic equation $Ax^2 + Bx + C = 0$, with $A \neq 0$, when the coefficients are independent, identically distributed uniform variates. The exact value of the roots can be obtained from the coefficients and the discriminant indicates if the roots are real or imaginary. Here, we consider the uniform distribution $U(-\theta, \theta)$ and find the probability of obtaining a real root to be 62.7%. This is done through simplification of the problem, analysis of the probability distribution of $B^2$ for both $U(-1, 1)$ and $U(0, 1)$, and final evaluation using conditional probability. Calculations are simplified by the fact that $B^2 \geq 4AC$ is always true when $AC \leq 0$. We leverage on the fact that the probability of obtaining real roots when coefficients are sampled from $U(0, \theta)$ is 25.4%. We verify the result experimentally through Monte Carlo simulation and present the desired supporting data accordingly.

Publication Details

Authors

Keywords

Quadratic equation, continuous uniform distribution, probability, Monte Carlo simulation

Follow-up Work

This paper laid the groundwork for a more general result. In a subsequent paper, we extended the analysis to the case where coefficients come from $U(\alpha, \beta)$ for any real numbers $\alpha$ and $\beta$, removing the symmetry constraint. That work was published in the Journal of the Indian Society for Probability and Statistics in 2023.

Access the Publication

The full paper is available as open access through the DOI link above. It contains the complete mathematical derivations, proofs, and simulation harness.

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