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will have real roots when the discriminant of the quadratic,
, is non-negative, i.e.
Your question about probability is currently impossible to answer without knowing exactly what the experiment is. Are you picking at random from some interval? Is the choice of either distributed a certain way?
I'll assume the inclusion of "[0,1]" in your question is a suggestion that both are chosen indepently of one another from [0, 1]. Let
denote the random variables that take on the values of
, respectively. I'll assume
are identical and follow the standard uniform distribution, i.e. they each have the same PDF and CDF as below:
where is either of
.
Then the question is to find . We have
and we can condition the random variable on the event of
by supposing
then integrate over all possible values of .