  Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be
the probability that a random set of $3$ vertices of $G$ spans a transitive
triangle (respectively an independent set). We prove that $t(G) + i(G) \geq
\frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we
supplement with several steps that make it more easily comprehensible. We also
prove a stability result and an exact result. Namely, we describe an extremal
construction, prove that it is essentially unique, and prove that if $H$ is
sufficiently far from that construction, then $t(H) + i(H)$ is significantly
larger than $\frac{1}{9}$. We go to greater technical detail than is usually
done in papers that rely on flag algebras. Our hope is that as a result this
text can serve others as a useful introduction to this powerful and beautiful
method.
