Molecular Ethology, An Immodest Proposal for Semantic Clarification - Understanding Understanding: Essays on Cybernetics and Cognition - page 151

FIGURE 5. Probability for drawing a white ball at the n th trial from an urn having
initially four balls of which 1, 2, or 3 are white, the others black. White balls are
replaced, black are not (a). Entropy at the n th trial (b).

which is the solution of

dp
dn

( )

= q 1 pp - (43)

or, recursively expressed, of

() =- ( ) + ( ) ◊ ( )

pn pn 1 q pn - 1 qn - 1 (44)

Figure 5a compares the probabilities p ( n ) for drawing a white ball at
the n th trial, as calculated through approximation [Eq. (42)] (solid curves),
with the exact values computed by an IBM 360/50 system with a pro-
gram kindly supplied by Mr. Atwood for an urn with initially four balls
( N = 4) and for the three cases in which one, two, or three of these are
white ( N w = 1, N w = 2, N w = 3). The entropy* H ( n ) in bits per trial corre-
sponding to these cases is shown in Fig. 5b, and one may note that while
for some cases [ p (1) £ 0.5] it reaches a maximum in the course of this game,

* Or the “amount of uncertainty”, or the “amount of information” received by the
outcome of each trial, defined by -H( n ) = p ( n )log 2 p ( n ) + q ( n )log 2 q ( n ).