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For best results, we recommend upgrading to the latest version of Internet Explorer or. Enter Password For The Encrypted File Setup Plant Design Suite 2011 keyQ:
What is the probability that the coordinates of a node in a random graph are smaller than the coordinates of the most external node?
I’m reading a problem in which a random graph is shown (figure).
I guess I need to calculate the area under the Venn Diagram, which will be the number of colored nodes (blue and red in this case), since the probability that I need to calculate is:
$$ \frac{n}{(n-1)^2} * (1-\frac{1}{(n-1)^2}) $$
However, this value is greater than 1, but in the answer, it says that the number of blue nodes is greater than the number of red nodes, which means that the blue node is greater than the red node (since they’re half of the nodes).
Could you please explain what is wrong in my reasoning?
A:
Please note that the number of red nodes and blue nodes are equal in this problem since
$$\frac{n}{(n-1)^2} = \frac{n-1}{(n-2)^2} = \frac{1}{(n-2)^2} = \frac{2}{n(n-1)} \ge \frac{1}{n-1}$$
Also, the number of red nodes in this problem is the number of nodes in the shaded red set, which is $\binom{n-2}{2}$, but you made a mistake by assigning $n$ to $(n-2)$ twice in your numerator. The correct way to calculate the shaded red set would be $n-2$ and $n-3$ of the vertices on the shaded red node since the red node has degree $2$.
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