- PROBABILITY 519
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23FIGURE 2. Histogram of the frequencies of the frequencies in
Fig. 1, compared with a normal distribution.Freq. (^6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)
Pred. (^0 0 1 1 2 3 4 4 5 5 4 4 3 2 2 1 1 0)
Obs. 1 1 0 (^1 2 2 3 4 4 5 3 8 2 3 1 1 0 1)
The agreement is not perfect, nor would we expect it to be. But it is remarkably
close, except for the one "outlier" at a frequency of 17, attained by eight numbers
instead of the theoretically predicted four. The mean for this model is 104/7 «
14.85, and the standard deviation is \/624/7 ~ 3.569. The histogram for the
frequencies of the frequencies, compared with the graph of the standard bell-shaped
curve with this mean and standard deviation are shown in Fig. 2. The fact that
mere numerical reasoning compels even the most chaotic phenomena to exhibit
some kind of order is one of the most awe-inspiring aspects of applied probability
theory. It is the phenomenon that led the British mathematician Francis Galton
(1822-1911) to describe the normal distribution as "the supreme law of unreason."
The Petersburg paradox. Soon after its introduction by Huygens and Jakob Bernoulli
the concept of mathematical expectation came in for some critical appraisal. While
working in the Russian Academy of Sciences, Daniel Bernoulli discussed the prob-
lem now known as the Petersburg paradox with his brother Niklaus (1695-1726,
known as Niklaus II). We can describe this paradox informally as follows. Suppose
that you toss a coin until heads appears. If it appears on the first toss, you win $2,
if it first appears on the second toss, you win $4, and so on; if heads first appears on
the nth toss, you win 2™ dollars. How much money would you be willing to pay to
play this game? Now by "rational" computations the expected winning is infinite,
being 2-|+4-| + 8| + -- -,so that you should be willing to pay, say, $10,000
to play each time. On the other hand, who would bet $10,000 knowing that there
was an even chance of winning back only $2, and that the odds are 7 to 1 against
winning more than $10? Something more than mere expectation was involved here.
Daniel Bernoulli discussed the matter at length in an article in the Comentarii of
the Petersburg Academy for 1730-1731 (published in 1738). He argued for the im-
portance of something that we now call utility. If you already possess an amount of
money ÷ and you receive a small additional amount of money dx, how much utility
does the additional money have for you, subjectively? Bernoulli assumed that the
increment of utility dy was directly proportional to dx and inversely proportional
to x, so that
kdx