Question

What is the probability that an E(3,1/3) random graph is not connected

Answer 1

Any graph with at most 22 edges is disconnected, any graph with at least 44 edges is connected (since there are only 33 edges among any 33 of the vertices). Thus you can do most of the work by summing probabilities for numbers of edges without dealing with specific configurations. Then you just have to consider which configurations of 33 edges make the graph connected and count them.

More explicit solution in response to the comments:

The probability that there are kk edges is P(K=k)=2−6(6k)P(K=k)=2−6(6k). You can either add these probabilities for kk from 44 to 66, or use the symmetry P(K<3)=P(K>3)P(K<3)=P(K>3) to get

P(K>3)=1−P(K=3)2=1−20642=1132.P(K>3)=1−P(K=3)2=1−20642=1132.

To this you need to add the probability of getting a connected graph with 33 edges. You correctly counted 1616 connected cases ((63)=20(63)=20 cases minus the 44 in which all 33 edges are among 33 vertices), so this probability is 16641664, for a total of

1132+1664=1932.