Sec. 1] THE RISK ELEMENT 267 
equally probable because the longer the trial is continued 
the more will the two tend toward equality. But they 
argue in a circle. It is not necessarily true that the 
longer the run the more closely will the frequency of the 
event approach its probability. For example, it is possible 
that though heads and tails have an equal chance, a run 
of heads may keep up for any given number of times, 
however long, a million, for instance; or that at first heads 
and tails may occur with equal frequency and as the ex- 
periment proceeds they may diverge more and more from 
such equality. No student of chance, whatever his theory 
of the philosophy of chance, would claim that these cases 
are vmpossible. The most that can be said is that they are 
extremely tmprobable. The statement, therefore, that the 
longer the run the more closely will the frequency of the 
event approach its probability turns out to be “the longer 
the run the more probably will the frequency correspond 
to the probability.” This is true as a proposition and it is 
in fact known as “Bernoulli’s Theorem ”’; but it cannot 
be made the basis of a sound definition of probability, for 
probability would be defined in terms of itself. It states 
that the probability of heads coming up is the frequency 
which heads will probably approximate in the long run! 
How else than in terms of probability can we formulate 
the conditions under which in the long run the coin 
“will ” fall according to its probability ? It is precisely 
at this point that the radical difficulty with the “long- 
run’’ theory is seen. It is said that in an athletic contest, 
the chance of winning is one half when two wrestlers are 
so nearly mated that in the long run “under precisely the 
same conditions,” each will win in half the contests. If 
the conditions are, literally speaking, precisely the same, 
then the same result will necessarily follow and the same 
man will always win. It is only as the conditions vary 
slightly from time to time in their unknown elements that 
there is a change of winner; and the instant the unknown-