ProbabilityHardBloom L3
Question
Suppose $m,n$ are real numbers randomly chosen in $\lbrack 0,1\rbrack$. Determine the probability that the distance between the roots of the equation $x^{2} + mx + n = 0$ is not greater than 1 .
Options
A.$\frac{2}{3}$
B.$\frac{1}{6}$
C.$\frac{1}{3}$
D.$\frac{1}{2}$
Solution
Sol. [Key Concept: Area probability for $m,n \in \lbrack 0,1\rbrack$ under the stated inequality.]
$$\begin{matrix} \left| x_{1} - x_{2} \right| & \ = \sqrt{\left( x_{1} + x_{2} \right)^{2} - 4x_{1}x_{2}} = \sqrt{m^{2} - 4n} \leq 1 \\ \left| {\text{ }m}^{2} - 4n \right| & \ \leq 1 \\ \Rightarrow - 1 \leq x^{2} - 4y & \ \leq 1 \end{matrix}$$
$$\begin{matrix} & A = \int_{0}^{1}\mspace{2mu}\mspace{2mu}\frac{x^{2} + 1}{4}dx = \frac{1}{3} \\ & P = \frac{1/3}{1} = \frac{1}{3} \end{matrix}$$
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