examday

/TMUA

Two parabolas have equations $f(x)=x^2+px+q$ and $g(x)=x^2+rx+s$, where $p,q,r$ and $s$ are integers each chosen independently and uniformly at random from 1 to 6 inclusive.

Consider the following four statements.

• I $p\ne r$ is necessary and sufficient for the parabolas to intersect at exactly one point.
• II The probability that the parabolas have at least one point in common is $\dfrac{31}{36}$.
• III The probability that the two parabolas are identical is $\dfrac1{36}$.
• IV If $p=r$, then the two parabolas have the same vertex.