Computing the probability of a certain complicated pattern occurring, after seeing it occur, is a tricky business. The chance of my mother marrying my father was one in two billions or so; that does not mean that my mere existence is a sign that something fishy is going one with the universe...
You said you read the upthread discussion, yet you continue the strawman. My point was..
The probability that any specific mother marries any specific father is in most cases quite rare, i.e. the independent trials are somewhat more uniformly distributed (they are all mostly rare, not some rare and others probable), thus the
frequently repeated occurrence of these roughly equivalently rare birth events is not rare.
You ignored my point that each independent coin toss trial outcome is
uniformly distributed whereas the Poisson distribution is
exponentially distributed.
That is why I asserted that your and xulescu's analogies are inapplicable. Rare trial outcomes in a Poisson distribution occur less often then less rare ones (look at the area under the distribution curve at the tails). Whereas all trial outcomes in a coin toss occur at the same probability.