Value of $a$ for real roots and finding who has a higher chance of success when one picks $a$ from $N(0,1)$ and another from an $U$?

Question

The quadratic is,$$x^2+ax+\ln \ 2=0$$

Person 1 draws for $a$ from $N(0,1)$ distribution.

Person 2 draws for $a$ from $ U(-2 \sqrt{2\pi},2 \sqrt{2\pi}) $

We say a person is successful if he draws a value of $a$ such that the quadratic has real roots. Who has a higher probability of being successful ?

My approach,

1. Found values of $a$ for which roots are real. Its coming out to be $(-\infty,-2 \sqrt{\ln2})\ \ \cup \ \ (2 \sqrt{\ln2},\infty)$

I dont know how to proceed ahead of this.

Answer 1

HINT: for the uniform distribution, you can easily determine the probability that a number $a$ randomly chosen in the range from $-2\sqrt{2\pi}$ and $-2\sqrt{2\pi}$ has absolute value $\geq 2\sqrt{\log 2}$, thus falling in the intervals that you have correctly identified. Simply calculate the proportion of the whole range that is identified by such intervals.

For the standard normal distribution, even without calculating the exact probability to get a number $a$ that satisfies the conditions above, you can remind the proportions of the values that are included within $1,2,3...$ standard deviations of the mean. This is sufficient to solve the problem.