tion without
first taking into account the number of families in the different
streets. If one street showed on the average ten times as many cases
as another, the coincidence might still be judged casual if there were
ten times as many families in it.

Apart from the fallacy of overlooking the positive frequency, certain
other fallacies or liabilities to error in applying this doctrine of
chances may be specified.

1. We are apt, under the influence of prepossession or prejudice, to
remember certain coincidences better than others, and so to imagine
extra-casual coincidence where none exists. This bias works in
confirming all kinds of established beliefs, superstitious and other,
beliefs in dreams, omens, retributions, telepathic communications, and
so forth. Many people believe that nobody who thwarts them ever comes
to good, and can produce numerous instances from experience in support
of this belief.

2. We are apt, after proving that there is a residuum beyond what
chance will account for on due allowance made for positive frequency,
to take for granted that we have proved some particular cause for
this residuum. Now we have not really explained the residuum by the
application of the principle of chances: we have only isolated a
problem for explanation. There may be more than chance will account
for: yet the cause may not be the cause that we assign off-hand. Take,
for example, the coincidence that has been remarked between race and
different forms of Christianity in Europe. If the distribution of
religious systems were entirely independent of race, it might be
said that you would expect one system to coincide equally often
with different races in proportion to the positive number of their
communities. But the Greek system is found almost solely among
Slavonic peoples, the Roman among Celtic, and the Protestant among
Teutonic. The coincidence is greater than chance will account for. Is
the explanation then to be found in some special adaptability of the
religious system to the character of the people? This may be the right
explanation, but we have not proved it by merely discounting chance.
To prove this we must show that there was no other cause at work, that
character was the only operative condition in the choice of system,
that political combinations, for example, had nothing to do with it.
The presumption from extra-casual coincidence is only that there is
a special cause: in determining what that is we mu