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\textit{II.  An Argument for Divine Providence, taken from the Constant
Regularity observed in the Births of both Sexes.  By} Dr. John
Arbuthnot, \textit{Physician in Ordinary to her Majesty, and
Fellow of the College of Physicians and the Royal Society.}
\footnote{From: \textit{Philosophical Transactions of the Royal Society of
London} \textbf{27} (1710), 186--190, reprinted in M~G~Kendall and
R~L~Plackett (eds), \textit{Studies in the History of Statistics and
Probability Volume II}, High Wycombe: Griffin 1977, pp.\,30--34.}

\bigskip

AMONG innumerable Footsteps of Divine Providence to be found in the
Works of Nature, there is a very remarkable one in the exact Ballance
that is  maintained between the Numbers of Men and Women; for by this
means it is provided, that the Species may never fail, nor perish, since
every Male may have its Female, and of a proportional Age. This Equality
of Males and Females is not the Effect of chance but Divine Providence,
working for a good End, which I thus demonstrate :

Let there be a Die of Two sides, M and F, which denote Cross and Pile),
now to find all the Chances of any determinate Number of such Dice, let
the Binome $\M + \F$ be raised to the Power, whose Exponent is the Number of
Dice given; the Coefficients of the Terms will show all the Chances
sought. For Example, in Two Dice of Two sides, M + F the chances are
$\M^2 + 2\M\F + \F^2$, that is One Chance for M double, One for F
double, and Two for M single and F single; in Four such Dice there are
Chances $\M^4 + 4\M^3\F + 6\M^2\F^2 + 4\M\F^3 + \F^4$; that is, One
Chance for M quadruple, One for F quadruple, Four for triple M and
single F,Four for single M and triple F. and Six for M double: and F
double: and universally,:  if the Number of Dice be $n$, all their
Chances will be expressed in this Series,\newline
$\M^n + \frac{\text{n}}{1}\times\M^{\text{n}-1}\F + \frac{\text{n}}{1}\times\frac{\text{n}-1}{2}\times\M^{\text{n}-2}\F^2 + \frac{\text{n}}{1}\times\frac{\text{n}-1}{2}\times\frac{\text{n}-2}{3} \times\M^{\text{n}-2}\F^2 +,\&c.$\newline
It appears. plainly, that when the Number of Dice is even, there are as
many M's as F's, in the middle Term of this Series, and in all the other
Terms there are most M's or most F's.

If therefore a Man undertake, with an even Number of Dice, to throw as
many M's as F's, he has all the Terms but the middle Term against him;
and his lot is the Sum of all the Chances, as the coefficient of the
middle Term, is to the power of 2 raised to an exponent equal to the
number of Dice: so in Two Dice, his Lot is $\frac{2}{4}$ or
$\frac{1}{2}$, in Three Dice $\frac{6}{16}$ or $\frac{3}{8}$, in Six
Dice $\frac{20}{64}$ or $\frac{5}{16}$, in Eight Dice $\frac{70}{256}$
or $\frac{35}{128}$, \&c.

To find this middle Term in any given Power or Number
of Dice, continue the Series
$\frac{\text{n}}{1}\times\frac{\text{n}-1}{2}\times\frac{\text{n}-2}{3},$
\&c.\ till the number of terms are equal to $\frac{1}{2}$n.  For
Example, the coefficient of the middle Term of the tenth Power
$\frac{10}{1}\times\frac{9}{2}\times{8}{3}\times\frac{7}{4}\times\frac{6}{5} =252$, the tenth Power of Two is 1024, if therefore A undertake to
throw with Ten Dice in one throw an equal Number of M's and F's,
he has 252 chances out of 1024 for him, that is his Lot is
$\frac{259}{1024}$ or $\frac{63}{256}$, which is less than $\frac{1}{2}$.

It will be easy by the help of Logarithms, to extend this Calculation to
a very great Number, but that is not my present Design.  It is visible
from what has been said, that with a very great Number of Dice, A's Lot
would become very small; and consequently (supposing  M to denote Male
and F Female) that in the vast Number of  Mortals, where would be but a
small part of all the possible Chances,  for its happening at any
assignable time, that an equal Number of Males and Females should be
born.

It is indeed to be confessed that this Equality of Males and Females is
not Mathematical but Physical, which alters much the foregoing
Calculation; for in this Case the middle Term will not exactly give A's
Chances, but his chances will take in some of the Terms next the middle
one, and will lean to one side or the other. But it is very improbable
(if mere Chance govern'd) that they would never reach as far as the
Extremities: But this Event is happily prevented by the wise Oeconomy of
Nature; and to judge of the wisdom of the Contrivance, we must observe
that the external Accidents to which Males are subject (who must seek.
their Food with danger) make a great havock of them, and that this loss
exceeds far that of the other Sex occasioned by Diseases incident to it,
as Experience convinces us.  To repair that Loss, provident Nature, by
the Disposal of its wise Creator, brings forth more Males than Females;
and that in almost a constant proportion. This appears from the annexed
Tables, which contain Observations for 82 years of the births in
\textit{London}. Now, to reduce the Whole to a Calculation, I propose
this

\textit{Problem}. A lays against B. that every Year there shall be born
more Males than Females: To find A's Lot, or the Value of his
Expectation.

It is evident from what has been said, that A's lot for each year is
less than $\frac{1}{2}$ (but, that the Argument might be stronger) let
his Lot be equal to $\frac{1}{2}$ for one year. If he undertakes to do the
same thing 82 times running, his Lot will be $\overline{\frac{1}{2}}|^{82}$,
which will be easily found by the Table of Logarithms to be
$\frac{1}{4\ 8360\ 0000\ 00000\ 00000\ 00000\ 0000}$.  But if A wager with
B, not only that the Number of Males shall exceed that of Females, every
Year, but that this Excess shall happen in a constant Proportion, and
the Difference lie within  fix'd limits; and this not only for 82 Years,
but for Ages of Ages, and not only at \textit{London}, but all over the
World; which it is highly probable is the Fact, and designed that every
Male may have a Female of the same Country and suitable Age; then A's
Chance will be near an infinitely small Quantity, at least less than any
assignable fraction.  From whence it flows, that it is Art, not Chance,
that governs.

There seems no more probable Cause to be assigned in Physics for
this Equality of the Births, than that in our 'first Parents
Seed there were at first formed an equal Number of both Sexes.

\textit{Scholium}. From hence it follows, that Polygamy is contrary to
the Law of Nature and Justice, and to the Propagation of the Human Race;
for where Males and Females are in equal number, if one \textit{M}an
take Twenty Wives, Nineteen Men must live in Celibacy, which is repugnant to
the Design of Nature; nor is it probable that Twenty Women will be so well
impregnated by one Man as by Twenty.

\begin{center}
\begin{tabular}{r|r|r||r|r|r}
\multicolumn{3}{c||}{Christened.} & \multicolumn{3}{c}{Christened.} \\
\textit{Anno.} & \textit{Males.} & \textit{Females.} &
\textit{Anno.} & \textit{Males.} & \textit{Females.} \\
1629 & 5218 & 4683 & 1648 & 3363 & 3181 \\
30 & 4858 & 4457 &   49 & 3079 & 2746 \\
31 & 4422 & 4102 &   50 & 2890 & 2722 \\
32 & 4994 & 4590 &   51 & 3231 & 2840 \\
33 & 5158 & 4839 &   52 & 3220 & 2908 \\
34 & 5035 & 4820 &   53 & 3196 & 2959 \\
35 & 5106 & 4928 &   54 & 3441 & 3179 \\
36 & 4917 & 4605 &   55 & 3655 & 3349 \\
37 & 4703 & 4457 &   56 & 3668 & 3382 \\
38 & 5359 & 4952 &   57 & 3396 & 3289 \\
39 & 5366 & 4784 &   58 & 3157 & 3013 \\
40 & 5518 & 5332 &   59 & 3209 & 2781 \\
41 & 5470 & 5200 &   60 & 3724 & 3247 \\
42 & 5460 & 4910 &   61 & 4748 & 4107 \\
43 & 4793 & 4617 &   62 & 5216 & 4803 \\
44 & 4107 & 3997 &   63 & 5411 & 4881 \\
45 & 4047 & 3919 &   64 & 6041 & 5681 \\
46 & 3768 & 3536 &   65 & 5114 & 4858 \\
47 & 3796 & 3536 &   66 & 4678 & 4319
\end{tabular}

\bigskip

\begin{tabular}{r|r|r||r|r|r}
\multicolumn{3}{c}{Christened.} & \multicolumn{3}{c}{Christened.} \\
\textit{Anno.} & \textit{Males.} & \textit{Females.} &
\textit{Anno.} & \textit{Males.} & \textit{Females.} \\
1667 & 5616 & 5322 & 1689 & 7604 & 7267 \\
68 & 6073 & 5560 &   90 & 7909 & 7302 \\
69 & 6506 & 5829 &   91 & 7662 & 7392 \\
70 & 6278 & 5719 &   92 & 7602 & 7316 \\
71 & 6449 & 6061 &   93 & 7676 & 7483 \\
72 & 6443 & 6120 &   94 & 6985 & 6647 \\
73 & 6073 & 5822 &   95 & 7263 & 6713 \\
74 & 6113 & 5738 &   96 & 7632 & 7229 \\
75 & 6058 & 5717 &   97 & 8062 & 7767 \\
76 & 6552 & 5847 &   98 & 8426 & 7626 \\
77 & 6423 & 6203 &   99 & 7911 & 7452 \\
78 & 6568 & 6033 & 1700 & 7578 & 7061 \\
79 & 6247 & 6041 & 1701 & 8102 & 7514 \\
80 & 6548 & 6299 & 1702 & 8031 & 7656 \\
81 & 6822 & 6533 & 1703 & 7765 & 7683 \\
82 & 6909 & 6744 & 1704 & 6113 & 5738 \\
83 & 7577 & 7158 & 1705 & 8366 & 7779 \\
84 & 7575 & 7127 & 1706 & 7952 & 7417 \\
85 & 7484 & 7246 & 1707 & 8239 & 7623 \\
86 & 7575 & 7119 & 1708 & 8239 & 7623 \\
87 & 7737 & 7214 & 1709 & 7840 & 7380 \\
88 & 7487 & 7101 & 1710 & 7640 & 7288
\end{tabular}
\end{center}

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