XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 293 
independence, of the Ng failures of the first event (Ng)g will be 
associated (on an average) with failures of the second event, and 
(&g)p with successes of the second event (cf. row 2 of the scheme 
on p. 292). Similarly of the Ap successful first events, (¥p)g will 
be associated (on an average) with failures of the second event 
and (Np)p with successes. In trials of two events we would 
therefore expect approximately Ng? cases of no success, 2Npg 
cases of one success and one failure, and Np? cases of two successes, 
as in row 3 of the scheme. The results of a third event may be 
combined with those of the first two in precisely the same way. 
Of the Ng® cases in which both the first two events failed, (¥q2)q 
will be associated (on an average) with failure of the third also, 
(N¢®)p with success of the third. Of the 2/Npq cases of one 
success and one failure, (28pg)g will be associated with failure 
of the third event and (28pg)p with success, and similarly for 
the Np? cases in which both the first two events succeeded. The 
result is that in A trials of three events we should expect Ng? 
cases of no success, 3 Npg? cases of one success, 3 Np? cases of two 
successes, and Np? cases of three successes, as in row 5 of the 
scheme. The scheme is continued for the results of a fourth 
event, and it is evident that all the results are included under a 
very simple rule: the frequencies of 0, 1, 2 . . . . successes are 
given 
for one event by the binomial expansion of N(g +p) 
for two events » ” Ng +p)? 
for three events i N(g+p)? 
for four events . fs N(g+p)t 
and soon. Quite generally, in fact :—the Jrequenciesof0,1,2 . . .. 
successes in IN trials of n events are given by the successive terms 
wn the binomial expansion of N(q +p)", viz.— 
n(n —1 n(n—1)(n-2 
vy "+ n.g" p+ Lon ) pry ( RS Jolson l 
This is the first theoretical expression that we have obtained for 
the form of a frequency-distribution, 
3. The general form of the distributions given by such 
binomial series will have been evident from the experimental 
examples given in Chapter XIII, i.e. they are distributions 
of greater or less asymmetry, tailing off in either direction 
from the mode. The distribution is, however, of so much 
importance that it is worth while considering the form in 
greater detail. This form evidently depends (1) on the values 
of ¢ and p, (2) on the value of the exponent n. If p and ¢ 
are equal, evidently the distribution must be symmetrical, for