I am trying to find the probability of a bernoulli random graph on $n=10$ vertices with probability that an edge connects any pair of vertices is $p=\frac{1}{6}$ as $n\to \infty$.
This is what I have:
$\lim_{n\to \infty}$ P($G$ contains $K_7$) = $\lim_{n\to \infty}$ P(at least $7$ vertices are all connected to each other)
= $1 -lim_{n\to \infty}$ P(at most $6$ vertices are all connected to each other)
= $1 -lim_{n\to \infty}$ P($2$) + P($3$) + P($4$) + p($5$) + P($6$)
My expression for the probability is where I think I go wrong. Here is just one part of it:
P($6) = {n\choose 6}\cdot p^{\dfrac{6\cdot 5}{2}}$
The reasoning is that there are ${n\choose 6}$ distinct subsets of size $6$ out of $n$ possible vertices. Then there are $\dfrac{6(6-1)}{2}$ edges in $K_6$. Since each edge has probability $p$ of occurring and each occurrence is independent of the others by definition of bernoulli random graph we raise $p$ to the $\dfrac{6\cdot 5}{2}$ power. Furthermore each distinct subset is independent and so we multiply this by ${n\choose 6}$
What must be wrong is the fact that the expression for P($6$) will be infinite since $p^{\dfrac{6\cdot 5}{2}}$ is a finite term.
- Where exactly did I go wrong?
- How can I improve my reasoning in combinatorics/probability?