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Jun 6, 2019 at 23:51 comment added Dan Romik This is quite a nice argument, but based on what OP wrote I have a feeling that they were asking for a “symmetry argument” that would directly show that the event in question has the same probability as its complement by somehow mapping them to each other. The symmetry you used is of a different sort, involving the fact that all edges of $G_n$ are equally likely to be in $T_n$.
Jun 6, 2019 at 5:10 history answered Yuval Peres CC BY-SA 4.0