Probability of triangles in $G(n, 1/2)$ I am trying to solve the following problem: Let $G = G(n, \frac{1}{2})$ be a random graph on $n$ vertices, i.e., for each pair of vertices $i, j$, we add the edge $(i, j)$ independently with probability $\frac{1}{2}$. Let $T_n$ be the number of triangles (i.e., cycles of length 3) in $G$. I am trying to find $E[T_n]$, the expectation of $T_n$, and $Var[T_n]$, the variance of $T_n$. Then, I need to find a way to prove that with high probability every vertex of $G$ is incident to a triangle (or in other words, let $p_n$ be the probability that for every vertex $i$, there is a triangle in $G$ containing $i$, and then prove that $p_n$ tends to 1 as $n$ tends to infinity.) Any help or suggestions would be appreciated! Thanks!

Hint: I would start by considering a single vertex $v$. 
If we knew how many neighbors $v$ has (degree of $v$), then how many triangles does it participate in? It is a random variable, but what is its conditioned on that degree?
Of course, we do not know the degree of $v$. But again what is the degree distribution?
If we knew the expected number of triangles for a vertex $v$, then what is the expected number of triangles in the entire graph? 
