> THEORY OF STATISTICS. 
dependent on the result of drawing the first. The disturbance 
can only be eliminated by drawing from a bag containing a 
number of balls that is infinitely large compared with the 
total number drawn, or by returning each ball to the bag before 
drawing the next. In this chapter our attention will be confined 
to the case of independent sampling, as in coin-tossing or dice- 
throwing—the simplest cases of an artificial kind suitable for 
theoretical study and experimental verification. For brevity, we 
may refer to such cases of sampling as simple sampling: the 
implied conditions are discussed more fully in § 8 below. 
4. If we may regard an ideal coin as a uniform, homogeneous 
circular disc, there is nothing which can make it tend to fall more 
often on the one side than on the other; we may expect, there- 
fore, that in any long series of throws the coin will fall with 
either face uppermost an approximately equal number of times, 
or with, say, heads uppermost approximately half the times. 
Similarly, if we may regard the ideal die as a perfect homogeneous 
cube, it will tend, in any long series of throws, to fall with each 
of its six faces uppermost an approximately equal number of 
times, or with any given face uppermost one-sixth of the whole 
number of times. These results are sometimes expressed by 
saying that the chance of throwing heads (or tails) with a coin is 
1/2, and the chance of throwing six (or any other face) with a die 
is 1/6. To avoid speaking of such particular instances as coins 
or dice, we shall in future, using terms which have become 
conventional, refer to an event the chance of success of which is p 
and the chance of failure ¢. Obviously p+¢=1. 
5. Suppose we take IV samples with » events in each. What 
will be the values towards which the mean and standard-deviation 
of the number of successes in a sample will tend? The mean is 
given at once, for there are N.n events, of which approximately 
pNn will be successes, and the mean number of successes in a 
sample will therefore tend towards pn. As regards the standard- 
deviation, consider first the single event (n=1). The single 
event may give either no successes or one success, and will tend 
to give the former gk, the latter pXV, times in XN trials. Take 
this frequency-distribution and work out the standard-deviation 
of the number of successes for the single event, as in the case of 
an arithmetical example :— 
Frequency f. Successes &. J 
2 g 
rN ' ] 
= y 
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