st conform to the
ordinary conditions of explanation.

So coincidence between membership of the Government and a classical
education may be greater than chance would account for, and yet the
circumstance of having been taught Latin and Greek at school may have
had no special influence in qualifying the members for their duties.
The proportion of classically educated in the Government may be
greater than the proportion of them in the House of Commons, and
yet their eminence may be in no way due to their education. Men of
a certain social position have an advantage in the competition for
office, and all those men have been taught Latin and Greek as a matter
of course. Technically speaking, the coinciding phenomena may be
independent effects of the same cause.

3. Where the alternative possibilities are very numerous, we are apt
not to make due allowance for the number, sometimes overrating it,
sometimes underrating it.

The fallacy of underrating the number is often seen in games of
chance, where the object is to create a vast number of alternatives,
all equally possible, equally open to the player, without his being
able to affect the advent of one more than another. In whist, for
example, there are some six billions of possible hands. Yet it is a
common impression that, one night with another, in the course of a
year, a player will have dealt to him about an equal number of good
and bad hands. This is a fallacy. A very much longer time is required
to exhaust the possible combinations. Suppose a player to have
2000 hands in the course of a year: this is only one "set," one
combination, out of thousands of millions of such sets possible. Among
those millions of sets, if there is nothing but chance in the matter,
there ought to be all proportions of good and bad, some sets all good,
some all bad, as well as some equally divided between good and bad.[1]

Sometimes, however, the number of possible alternatives is overrated.
Thus, visitors to London often remark that they never go there without
meeting somebody from their own locality, and they are surprised at
this as if they had the same chance of meeting their fellow-visitors
and any other of the four millions of the metropolis. But really the
possible alternatives of rencounter are far less numerous. The places
frequented by visitors to London are filled by much more limited
numbers: the possible rencounters are to be counted by thousands
rather than by millions.