Probability of a balanced triangle in graph

Question

I have a complete graph of $n$ nodes. All edges are undirected.

Now I mark each sign of edge $e$ positive with probability $p$ and negative with probability $1-p$.

I wonder whats the probability that a triangle of any 3 nodes in this graph is balanced. A triangle is balanced, if every edge in this triangle is $\ge 0$.

Can anyone help me? Thank you so much!

Answer 1

Since all nodes are connected you chose 3 nodes at random. So for the first edge of the three nodes the probability of being positive is $p$, as it is for the second and third.

So the total probability of having positive signs one all of the three edges is $p^3$.