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'Say the ratio of maniraptorans living in the K to those living in the J was
Say the ratio of the chances of an indiv living in the K ever being found to
that of one in the J is C:D;
The ratio of the chances of a maniraptoran we have found coming from the K
to from the J is (A/B)*(C/D).
The probability of any maniraptoran we have found coming from the K is  1/(1
+ 1/ ((A/B) * (C/D)) ).

The chance of N maniraptoran fossils all appearing after Archaeopteryx is
this expression raised to the power of N.'

It'll take me a while to untangle the parentheses and confirm what I think
you're saying.
However, remember that small proportions of small samples have huge ranges and
that any detail of the group you're looking at has approximately no validity
at all.
Awhile ago I was analyzing the results of a 1,000 response telephone sample
which found a less than 1% rate of probable pathological gamblers.  Guess how
valid the demographics of that subsample are.  To approach useful numbers
you'd need 6 or 7 times as many samples (at about $70 apiece) and
stratification (does not mean under rocks).
Aside from this issue, remember that fossils are found not only where they can
be preserved but also where they can be looked for comparatively easily.
If you wanted to be sure of getting a valid sample of a world population you'd
have to look in a lot of randomly selected places.  You might end up in my

In re birds, are you satisfied that the gap between archaeopteryx and the next
'bird' is short enough to disprove the assertion that archaeopteryx was an
anticipation of birds, but not necessarily even a close relative of the bird

Seems like this type analysis has rules for validating hypotheses I'm not sure
I understand.