Lii. An Essay Towards Solving A Problem In The Doctrine Of Chances ...

LII. An Essay towards solving a Problem in the Doctrine
of Chances. By the late Rev. Mr. Bayes, communicated
by Mr. Price, in a letter to John Canton, M. A. and
F. R. S.
Dear Sir,
Read Dec. 23, 1763. I now send you an essay which I have found among the papers
of our deceased friend Mr. Bayes, and which, in my opinion, has great merit,
and well deserves to be preserved. Experimental philosophy, you will ﬁnd, is
nearly interested in the subject of it; and on this account there seems to be
particular reason for thinking that a communication of it to the Royal Society
cannot be improper.
He had, you know, the honour of being a member of that illustrious So-
ciety, and was much esteemed by many as a very able mathematician. In an
introduction which he has writ to this Essay, he says, that his design at ﬁrst in
thinking on the subject of it was, to ﬁnd out a method by which we might judge
concerning the probability that an event has to happen, in given circumstances,
upon supposition that we know nothing concerning it but that, under the same
circumstances, it has happened a certain number of times, and failed a certain
other number of times. He adds, that he soon perceived that it would not be
very diﬃcult to do this, provided some rule could be found, according to which
we ought to estimate the chance that the probability for the happening of an
event perfectly unknown, should lie between any two named degrees of prob-
ability, antecedently to any experiments made about it; and that it appeared
to him that the rule must be to suppose the chance the same that it should
lie between any two equidiﬀerent degrees; which, if it were allowed, all the rest
might be easily calculated in the common method of proceeding in the doctrine
of chances. Accordingly, I ﬁnd among his papers a very ingenious solution of
this problem in this way. But he afterwards considered, that the postulate on
which he had argued might not perhaps be looked upon by all as reasonable;
and therefore he chose to lay down in another form the proposition in which he
thought the solution of the problem is contained, and in a Scholium to subjoin
the reasons why he thought it so, rather than to take into his mathematical
reasoning any thing that might admit dispute. This, you will observe, is the
method which he has pursued in this essay.
Every judicious person will be sensible that the problem now mentioned is by
no means merely a curious speculation in the doctrine of chances, but necessay
to be solved in order to a sure foundation for all our reasonings concerning past
facts, and what is likely to be hereafter. Common sense is indeed suﬃcient to
shew us that, form the observation of what has in former instances been the
consequence of a certain cause or action, one may make a judgement what is
likely to be the consequence of it another time. and that the larger number of
experiments we have to suypport a conclusion, so much more the reason we have
to take it for granted. But it is certain that we cannot determine, at least not to

any nicety, in what degree repeated experiments conﬁrm a conclusion, without
the particular discussion of the beforementioned problem; which, therefore, is
necessary to be considered by any that would give a clear account of the strength
of analogical or inductive reasoning; concerning, which at present, we seem to
know little more than that it does sometimes in fact convince us, and at other
times not; and that, as it is the means of cquainting us with many truths, of
which otherwise we must have been ignourant; so it is, in all probability, the
source of many errors, which perhaps might in some measure be avoided, if the
force that this sort of reasoning ought to have with us were more distinctly and
clearly understood.
These observations prove that the problem enquired after in this essay is no
less important than it is curious. It may be safely added, I fancy, that it is
also a problem that has never before been solved. Mr. De Moivre, indeed, the
great improver of this part of mathematics, has in his Laws of chance∗, after
Bernoulli, and to a greater degree of exactness, given rules to ﬁnd the probability
there is, that if a very great number of trials be made concerning any event,
the proportion of the number of times it will happen, to the number of times
it will fail in those trials, should diﬀer less than by small assigned limits from
the proportion of its failing in one single trial. But I know of no person who
has shown how to deduce the solution of the converse problem to this; namely,
“the number of times an unknown event has happened and failed being given,
to ﬁnd the chance that the probability of its happening should lie somewhere
between any two named degrees of probability.”
What Mr. De Moivre has
done therefore cannot be thought suﬃcient to make the consideration of this
point unnecessary: especially, as the rules he has given are not pretended to
be rigorously exact, except on supposition that the number of trials are made
inﬁnite; from whence it is not obvious how large the number of trials must be
in order to make them exact enough to be depended on in practice.
Mr. De Moivre calls the problem he has thus solved, the hardest that can
be proposed on the subject of chance. His solution he has applied to a very
important purpose, and thereby shewn that those a remuch mistaken who have
insinuated that the Doctrine of Chances in mathematics is of trivial consequence,
and cannot have a place in any serious enquiry†. The purpose I mean is, to shew
what reason we have for believing that there are in the constitution of things
ﬁxt laws according to which things happen, and that, therefore, the frame of
the world must be the eﬀect of the wisdom and power of an intelligent cause;
and thus to conﬁrm the argument taken from ﬁnal causes for the existence of
the Deity. It will be easy to see that the converse problem solved in this essay is
more directly applicable to this purpose; for it shews us, with distinctness and
precision, in every case of any particular order or recurrency of events, what
reason there is to think that such recurrency or order is derived from stable
causes or regulations innature, and not from any irregularities of chance.
∗See Mr. De Moivre’s Doctrine of Chances, p. 243, &c. He has omitted the demonstration
of his rules, but these have been supplied by Mr. Simpson at the conclusion of his treatise on
The Nature and Laws of Chance.
†See his Doctrine of Chances, p. 252, &c.
2

The two last rules in this essay are given without the deductions of them.
I have chosen to do this because these deductions, taking up a good deal of
room, would swell the essay too much; and also because these rules, though
not of considerable use, do not answer the purpose for which they are given
as perfectly as could be wished. They are however ready to be produced, if a
communication of them should be thought proper. I have in some places writ
short notes, and to the whole I have added an application of the rules in this
essay to some particular cases, in order to convey a clearer idea of the nature
of the problem, and to shew who far the solution of it has been carried.
I am sensible that your time is so much taken up that I cannot reasonably
expect that you should minutely examine every part of what I now send you.
Some of the calculations, particularly in the Appendix, no one can make without
a good deal of labour. I have taken so much care about them, that I believe
there can be no material error in any of them; but should there be any such
errors, I am the only person who ought to be considered as answerable for them.
Mr. Bayes has thought ﬁt to begin his work with a brief demonstration of the
general laws of chance. His reason for doing this, as he says in his introduction,
was not merely that his reader might not have the trouble of searching elsewhere
for the principles on which he has argued, but because he did not know whither
to refer him for a clear demonstration of them. He has also make an apology
for the peculiar deﬁnition he has given of the word chance or probability. His
design herein was to cut oﬀ all dispute about the meaning of the word, which in
common language is used in diﬀerent senses by persons of diﬀerent opinions, and
according as it is applied to past or future facts. But whatever diﬀerent senses it
may have, all (he observes) will allow that an expectation depending on the truth
of any past fact, or the happening of any future event, ought to be estimated so
much the more valuable as the fact is more likely to be true, or the event more
likely to happen. Instead therefore, of the proper sense of the word probability,
he has given that which all will allow to be its proper measure in every case
where the word is used. But it is time to conclude this letter. Experimental
philosophy is indebted to you for several discoveries and improvements; and,
therefore, I cannot help thinking that there is a peculiar propriety in directing
to you the following essay and appendix. That your enquiries may be rewarded
with many further successes, and that you may enjoy every valuable blessing,
is the sincere wish of, Sir,
your very humble servant,
Richard Price.
Newington Green,
Nov. 10, 1763.
3

P R O B L E M.
Given the number of times ion which an unknown event has happende and failed:
Required the chance that the probability of its happening in a single trial lies
somewhere between any two degrees of probability that can be named.
S E C T I O N
I.
DEFINITION 1. Several events are inconsistent, when if one of them hap-
pens, none of the rest can.
2. Two events are contrary when one, or other of them must; and both
together cannot happen.
3. An event is said to fail, when it cannot happen; or, which comes to the
same thing, when its contrary has happened.
4. An event is said to be determined when it has either happened or failed.
5. The probability of any event is the ratio between the value at which an
expectation depending on the happening of the event ought to be computed,
and the chance of the thing expected upon it’s happening.
6. By chance I mean the same as probability.
7. Events are independent when the happening of any one of them does
neither increase nor abate the probability of the rest.
P R O P.
1.
When several events are inconsistent the probability of the happening of one
or other of them is the sum of the probabilities of each of them.
Suppose there be three such events, and which ever of them happens I am
to receive N, and that the probability of the 1st, 2d, and 3d are respectively
a , b , c . Then (by deﬁnition of probability) the value of my expectation from
N
N
N
the 1st will be a, from the 2d b, and from the 3d c. Wherefore the value of
my expectations from all three is in this case an expectations from all three
will be a + b + c. But the sum of my expectations from all three is in this
case an expectation of receiving N upon the happening of one or other of them.
Wherefore (by deﬁnition 5) the probability of one or other of them is a+b+c or
N
a + b + c . The sum of the probabilities of each of them.
N
N
N
Corollary. If it be certain that one or other of the events must happen,
then a + b + c = N. For in this case all the expectations together amounting
to a certain expectation of receiving N, their values together must be equal to
N. And from hence it is plain that the probability of an event added to the
probability of its failure (or its contrary) is the ratio of equality. For these are
two inconsistent events, one of which necessarily happens. Wherefore if the
probability of an event is P that of it’s failure will be N−P .
N
N
P R O P.
2.
If a person has an expectation depending on the happening of an event, the
probability of the event is to the probability of its failure as his loss if it fails to
his gain if it happens.
4

Suppose a person has an expectation of receiving N, depending on an event
the probability of which is P . Then (by deﬁnition 5) the value of his expectation
N
is P, and therefore if the event fail, he loses that which in value is P; and if it
happens he receives N, but his expectation ceases. His gain therefore is N − P.
Likewise since the probability of the event is P , that of its failure (by corollary
N
prop. 1) is N−P . But N−P is to P as P is to N − P, i.e. the probability of the
N
N
N
event is to the probability of it’s failure, as his loss if it fails to his gain if it
happens.
P R O P.
3.
The probability that two subsequent events will both happen is a ratio com-
pounded of the probability of the 1st, and the probability of the 2d on suppo-
sition the 1st happens.
Suppose that, if both events happen, I am to receive N, that the probability
both will happen is P , that the 1st will is a (and consequently that the 1st will
N
N
not is N−a ) and that the 2d will happen upon supposition the 1st does is b .
N
N
Then (by deﬁnition 5) P will be the value of my expectation, which will become
b is the 1st happens. Consequently if the 1st happens, my gain is b − P, and if
it fails my loss is P. Wherefore, by the foregoing proposition, a is to N−a , i.e.
N
N
a is to N − a as P is to b − P. Wherefore (componendo inverse) a is to N as P
is to b. But the ratio of P to N is compounded of the ratio of P to b, and that
of b to N. Wherefore the same ratio of P to N is compounded of the ratio of a
to N and that of b to N, i.e. the probability that the two subsequent events will
both happen is compounded of the probability of the 1st and the probability of
the 2d on supposition the 1st happens.
Corollary. Hence if of two subsequent events the probability of the 1st be
a , and the probability of both together be P , then the probability of the 2d on
N
N
supposition the 1st happens is P .
a
P R O P.
4.
If there be two subesequent events be determined every day, and each day
the probability of the 2d is b and the probability of both P , and I am to receive
N
N
N if both of the events happen the 1st day on which the 2d does; I say, according
to these conditions, the probability of my obtaining N is P . For if not, let the
b
probability of my obtaining N be x and let y be to x as N − b to N. The since
N
x is the probability of my obtaining N (by deﬁnition 1) x is the value of my
N
expectation. And again, because according to the foregoing conditions the 1st
day I have an expectation of obtaining N depdening on the happening of both
events together, the probability of which is P , the value of this expectation is
N
P. Likewise, if this coincident should not happen I have an expectation of being
reinstated in my former circumstances, i.e. of receiving that which in value is x
depending on the failure of the 2d event the probability of which (by cor. prop.
1) is N−b or y , because y is to x as N − b to N. Wherefore since x is the thing
N
x
expected and y the probability of obtaining it, the value of this expectation is
x
5

y. But these two last expectation is y. But these two last expectations together
are evidently the same with my original expectastion, the value of which is x,
and therefor P + y = x. But y is to x as N − b is to N. Wherefore x is to P as
N is to b, and x (the probability of my obtaining N) is P .
N
b
Cor. Suppose after the expectation given me in the foregoing proposition,
and before it is at all known whether the 1st event has happened or not, I should
ﬁnd that the 2d event is determined on which my expectation depended, and
have no reason to esteem the value of my expectation either greater or less, it
would be reasonable for me to give something to be reinstated in my former
circumstances, and this over and over again as I should be informed that the 2d
event had happened, which is evidently absurd. And the like absurdity plainly
follows if you say I ought to set a greater value on my expectation than before,
for then it would be reasonable for me to refuse something if oﬀered me upon
condition I would relinquish it, and be reinstated in my former circumstances;
and this likewise over and over again as often as (nothing being known concern-
ing the 1st event) it should appear that the 2d had happened. Notwithstanding
therefore that the 2d event has happened, my expectation ought to be esteemed
the same as before i. e. x, and consequently the probability of my obtaining
N is (by deﬁnition 5) still x or P ∗. But after this discovery the probability
N
b
of my obtaining N is the probability that the 1st of two subsequent events has
happen=ed upon the supposition that the 2d has, whose probabilities were as
before speciﬁed. But the probability that an event has happened is the same as
as the probability I have to guess right if I guess it has happened. Wherefore
the following proposition is evident.
P R O P.
5.
If there be two subsequent events, the probability of the 2d b and the
N
probability of both together P , and it being 1st discovered that the 2d event
N
has slso happened, the probability I am right is P †.
b
∗What is here said may perhaps be a little illustrated by considering that all that can be
lost by the happening of the 2d event is the chance I should have of being reinstated in my
formed circumstances, if the event on which my expectation depended had been determined
in the manner expressed in the propostion. But this chance is always as much against me as
it is for me. If the 1st event happens, it is against me, and equalto the chance for the 2d
event’s failing. If the 1st event does not happen, it is for me, and equalalso to the chance for
the 2d event’s failing. The loss of it, therefore, can be no disadvantage.
†What is proved by Mr. Bayes in this and the preceding proposition is the same with
the answer to the following question. What is the probability that a certain event, when it
happens, will be accompanied with another to be determined at the same time? In this case,
as one of the events is given, nothing can be due for the expectation of it; and, consequently,
the value of an expectation depending on the happening of both events must be the same
with the value of an expectation depending on the happening of one of them.
In other
words; the probability that, when one of two events happens, the other will, is the same
with the probability of this other. Call x then the probability of this other, and if b be
N
the probability of the given event, and p the probability of both, because p = b × x,
N
N
N
x = p = the probability mentioned in these propositions.
b
6

P R O P.
6.
The probability that several independent events shall happen is a ratio com-
pounded of the probabilities of each.
For from the nature of independent events, the probability that any one
happens is not altered by the happening or gailing of any one of the rest, and
consequently the probability that the 2d event happens on supposition the 1st
does is the same with its original probability; but the probability that any two
events happen is a ratio compounded of the 1st event, and the probability of the
2d on the supposition on the 1st happens by prop. 3. Wherefore the probability
that any two independent events both happen is a ratio compounded of the 1st
and the probability of the 2d. And in the like manner considering the 1st and 2d
events together as one event; the probability that three independent events all
happen is a ratio compounded of the probability that the two 1st both happen
and the probability of the 3d. And thus you may proceed if there be ever so
many such events; from which the proposition is manifest.
Cor. 1. If there be several independent events, the probability that the 1st
happens the 2d fails, the 3d fails and the 4th happens, &c. is a ratio compounded
of the probability of the 1st, and the probability of the failure of 2d, and the
probability of the failure of the 3d, and the probability of the 4th, &c. For the
failure of an event may always be considered as the happening of its contrary.
Cor. 2. If there be several independent events, and the probability of each
one be a, and that of its failing be b, the probability that the 1st happens and
the 2d fails, and the 3d fails and the 4th happens, &c. will be abba, &c. For,
according to the algebraic way of notation, if a denote any ratio and b another
abba denotes the ratio compounded of the ratios a, b, b, a. This corollary is
therefore only a particular case of the foregoing.
Deﬁnition. If in consequence of certain data there arises a probability that
a certain event should happen, its happening or failing, in consequence of these
data, I call it’s happening or failing in the 1st trial. And if the same data be
again repeated, the happening or failing of the event in consequence of them I
call its happening or failing in the 2d trial; and so again as often as the same
data are repeated. And hence it is manifest that the happening or failing of the
same event in so many diﬀere- trials, is in reality the happening or failing of so
many distinct independent events exactly similar to each other.
P R O P.
7.
If the probability of an event be a, and that of its failure be b in each single
trial, the probability of its happening p times, and failing q times in p+q trials is
E apbq if E be the coeﬃcient of the term in which occurs apbq when the binomial
a + b|b+q is expanded.
For the happening or failing of an event if diﬀerent trials are so many inde-
pendent events. Wherefore (by cor. 2. prop. 6.) the probability that the event
happens the 1st trial, fails the 2d and 3d, and happens the 4th, fails the 5th.
&c. (thus happening and failing till the number of times it happens be p and the
7

number it fails be q) is abbab &c. till the number of a’s be p and the number of
b’s be q, that is; ’tis apbq. In like manner if you consider the event as happening
p times and failing q times in any other particular order, the probability for it
is apbq; but the number of diﬀerent orders according to which an event may
happen or fails so as in all to happen p times and fail q, in p + q trials is equal
to the number of permutations that aaaa bbb admit of when the number of a’s
is p and the number of b’s is q. And this number is equal to E, the coeﬃcient of
the term in which occurs apbq when a + b|p+q is expanded. The event therefore
may happen p times and fail q in p + q trials E diﬀerent ways and no more, and
its happening and failing these several diﬀerent ways are so many inconsistent
events, the probability for each of which is apbq, and therefore by prop. 1. the
probability that some way or other it happens p times and fails q times in p + q
trials is E apbq.
S E C T I O N
II.
Postulate. 1. Suppose the square table or plane ABCD to be so made and
levelled, that if either of the balls o or W be thrown upon it, there shall be the
same probability that it rests upon any one equal part of the plane as another,
and that it must necessarily rest somewhere upon it.
2. I suppose that the ball W shall be 1st thrown, and through the point
where it rests a line os shall be drawn parallel to AD, and meeting CD and AB
in s and o; and that afterwards the ball O shall be thrown p + q or n times, and
that its resting between AD and os after a single throw be called the happening
of the event M in a single trial. These things supposed,
Lem. 1. The probability that the point o will fall between any two points
in the line AB is the ratio of the distance between the two points to the whole
line AB.
Let any two points be named, as f and b in the line AB, and through them
parallel to AD draw f F , bL meeting CD in F and L. Then if the rectangles Cf ,
Fb, LA are commensurable to each other, they may each be divided into the
same equal parts, which being done, and the ball W thrown, the probability it
will rest somewhere upon any number of these equal parts will be the sum of the
probabilities it has to rest upon each one of them, because its resting upon any
diﬀerent parts of the plance AC are so many inconsistent events; and this sum,
because— the probability it should rest upon any one equal part as another is
the same, is the probability it should rest upon any one equal part multiplied
by the number of parts. Consequently, the probability there is that the ball
W should rest somewhere upon Fb is the probability it has to rest upon one
equal part multiplied by the number of equal parts in Fb; and the probability
it rests somewhere upon Cf or LA, i.e. that it dont rest upon Fb (because it
must rest somewhere upon AC) is the probability it rests upon one equal part
multiplied by the number of equal parts in Cf , LA taken together. Wherefore,
the probability it rests upon Fb is to the probability it dont as the number of
equal parts in Fb is to the number of equal parts in Cf , LA together, or as Fb
to Cf , LA together, or as f b to Bf , Ab together. And (compendo inverse) the
8

probability it rests upon Fb added to the probability it dont, as f b to A B, or
as the ratio of f b to AB to the ratio of AB to AB. But the probability of any
event added to the probability of its failure is the ratio of equality; wherefore,
the probability if rest upon Fb is to the ratio of equality as the ratio of f b to AB
to the ratio of AB to AB, or the ratio of equality; and therefore the probability
it rest upon Fb is the ratio of f b to AB. But ex hypothesi according as the ball
W falls upon Fb or nor the point o will lie between f and b or not, and therefore
the probability the point o will lie between f and b is the ratio of f b to AB.
Again; if the rectangles Cf , Fb, LA are not commensurable, yet the last
mentioned probability can be neither greater nor less than the ratio of f b to
AB; for, if it be less, let it be the ratio of f c to AB, and upon the line f b take
the points p and t, so that pt shall be greater than half cb, and taking p and t the
nearest points of division to f and c that lie upon f b). Then because Bp, pt, tA
are commensurable, so are the rectangles Cp, Dt, and that upon pt compleating
the square AB. Wherefore, by what has been said, the probability that the point
o will lie between p and t is the ratio of pt to AB. But if it lies between p and
t it must lie between f and b. Wherefore, the probability it should lie between
f and b cannot be less than the ratio of f c to AB (since pt is greater than f c).
And after the same manner you may prove that the forementioned probability
cannot be greater than the ratio of f b to AB, it must therefore be the same.
Lem. 2. The ball W having been thrown, and the line os drawn, the proba-
bility of the event M in a single trial is the ratio of Ao to AB.
For, in the same manner as in the foregoing lemma, the probability that the
ball o being thrown shall rest somewhere upon Do or between AD and so is the
ratio of Ao to AB. But the resting of the ball o between AD and so after a single
thrwo is the happening of the event M in a single trial. Wherefore the lemma
is manifest.
P R O P.
8.
If upon BA you erect the ﬁgure BghikmA whose property is this, that (the
base BA being divided into any two parts, as Ab, and Bb and at the point of
division b a perpendicular being erected and terminated by the ﬁgure in m; and
y, x, r representing respectively the ratio of bm, Ab, and Bb to AB, and E being
the coeﬃcient of the term which occurs in apbq when the binomial a + b|p+q is
expanded) y = Exprq. I say that before the ball W is thrown, the probability
the point o should fall between f and b, any two o=points named in the line AB,
and withall that the event M should happen p times and fail q in p + q trials, is
the ratio of f ghikmb, the part of the ﬁgure BghikmA intercepted between the
perpendiculars f g, bm raised upon the line AB, to CA the square upon AB.
D E M O N S T R A T I O N.
For if not; 1st let it be the ratio of D a ﬁgure greater than f ghikmb to CA,
and through the points e, d, c draw perpendiculars to f b meeting the curve
AmigB in h, i, k; the point d being so placed that di shall be the longest of the
perpendiculars terminated by the line f b, and the curve AmigB; and the points
9

e, d, c being so many and so placed that the rectangles bk, ci, ei, f b taken
together shall diﬀer less from f ghikmb than D does; all which may be easily
done by the help of the equation of the curve, and the diﬀerence between D
and the ﬁgure f ghikmb given. Then since di is the longest of the perpendicular
ordinates that insist upon f b, the rest will gradually decrease as they are farther
and farther from it on each side, as appears from the construction of the ﬁgure,
and consequently eb is greater than gf or any other ordinate that insists upon
ef .
Now if Ao were equal to Ae, then by lem. 2. the probability of the event M in
a single trial would be the ratio of Ae to AB, and consequently by cor. Prop. 1.
the probability of it’s failure would be the ratio of Be to AB. Wherefore, if x and
r be the two forementioned ratios respectively, by Prop. 7. the probability of the
event M happening p times and failing q in p + q trials would be E xprq . But x
and r being respectively the ratios of Ae to AB and Be to AB, if y is the ratio of
eb to AB, then, by construction of the ﬁgure AiB, y = Exprq. Wherefore, if Ao
were equal to Ae the probability of the event M happening p times and failing q
times in p + q trials would be y, or the ratio of eb to AB. And if Ao were equal to
Af , or were any mean between Ae and Af , the last mentioned probability for
the same reasons would be the ratio of f g or some other of the ordinates insisting
upon ef , to AB. But eh is the greatest of all the ordinates that insist upon ef .
Wherefore, upon supposition the point should lie any where between f and e,
the probability that the event M happens p times and fails q in p + q trials can’t
be greater than the ratio of eh to AB. There then being these two subsquent
events. the 1st that the point o will lie between e and f , the 2d that the event
M will happen p times and fail q in p + q trials, and the probability of the 1st
(by lemma 1st) is the ratio of ef to AB, and upon supposition the 1st happens,
by what has now been proved, the probability of the 2d cannot be greater than
the ratio of eh to AB it evidently follows (from Prop. 3.) that the probability
both together will happen cannot be greater than the ratio compounded of that
of ef to AB and that of eh to AB, which compound ratio is the ratio of f h to
CA. Wherefore, the probability that the point o will lie between f and e, and
the event M will happen p times and fqil q, is not greater than the raio of f h
to CA. And in like, manner the probability the point o will lie between e and
d, and the event M happen and fail as before, cannot be greater than the raio
of ei to CA. And again, the probability the point o will lie between c and b,
and the event M happen and fail as before, cannot be greater than the ratio of
bk to CA. Add now all these several probabilities together, and their sum (by
Prop. 1.) will be the probability that the point will lie somewhere between f
and b, and the event M happen p times and fail q in p + q trials. Add likewise
the correspondent ratios together, and their sum will be the ratio of the sum of
the antecedents to their consequent, i. e. the ratio of f b, ei, ci, bk together to
CA; which ratio is less than that of D to CA, because D is greater than f h, ei,
ci, bk together. And therefore, the probability that the point o will lie between
f and b, and withal that the event M will happen p times and fail q in p + q
times, is less than the ratio of D to CA; but it was supposed the same which
is absurd. And in like manner, by inscribing rectangles within the ﬁgure, as eg,
10

dh, dk, cm you may prove that the last mentioned probability is greater than
the ratio of any ﬁgure less than f ghikmb to CA.
Wherefore, that probability must be the ratio of f ghikmb to CA.
Cor. Before the ball W is thrown the probability that the point o will lie
somwehere between A and B, or somewhere upon the line AB, and withal that
the event M will happen p times, and fail q in p + q trials is the ratio of the
whole ﬁgure AiB to ZCA. But it is certain that the point o will lie somewhere
upon AB. Wherefore, before the ball W is thrown the probability the event M
will happen p times and fail q in p + q trials is the ratio of AiB to CA.
P R O P.
9.
If before any thing is discovered the place of the point o, it should appear
that the event M had happened p times and failed q in p + q trials, and from
hence I guess that the point o lies between any two points in the line AB, as f
and b, and consequently that the probability of the event M in a single trial was
somewhere between the ratio of Ab to AB and that of Af to AB: the probability
I am in the right is the ratio of that part of the ﬁgure AiB described as before
which is intercepted between perpendiculars erected upon AB at the points f
and b, to the whole ﬁgure AiB.
For, there being these two subsequent events. the ﬁrst that the point o will
lie between f and b; the second that the event M should happen p times and
fail q in p + q trials; and (by cor. prop. 8.) the original probability of the second
is the ratio of AiB to CA, and (by prop. 8.) the probability of both is the ratio
of f ghikmb to CA; wherefore (by prop. 5) it being ﬁrst discovered that the
second has happened, and from hence I guess that the ﬁrst has happened also,
the probability I am in the right is the ratio of f ghimb to AiB, the point which
was to be proved.
Cor. The same things supposed, I guess that the probability of the event M
lies somewhere between o and the ratio of Ab to AB, my chance to be in the
right is the ratio of Abm to AiB.
S c h o l i u m.
From the preceding proposition it is plain, that in the case of such an event
as I there call M, from the number of trials it happens and fails in a certain
number of trials, without knowing any thing more concerning it, one may give a
guess whereabouts it’s probability is, and, by the usual methods computing the
magnitudes of the areas there mentioned see the chance that the guess is right.
And that the same rule is the proper one to be used in the case of an event
concerning the probability of which we absolutely know nothing antecedently to
any trials made concerning it, seems to appear from the following consideration:
viz. that concerning such an event I have no reason to think that, in a certain
number of trials, it should rather happen any one possible number of times
than another. For, on this account, I may justly reason concerning it as if its
probability had been at ﬁrst unﬁxed, and then determined in such a manner
as to give me no reason to think that, in a certain number of trials, it should
11

rather happen any one possible number of times rather than another. But this
is exactly the case of the event M. For before the ball W is thrown, which
determines it’s probability in a single trial, (by cor. prop. 8.) the probability
it has to happen p times and fail q in p + q or n trials is the ratio of AiB to
CA, which ratio is the same when p + q or n is given, whatever number p is; as
will appear by computing the magnitude of AiB by the method ∗ of ﬂuxions.
And consequently before the place of the point o is discovered or the number of
times the event M has happened in n trials, I have not reason to think it should
rather happen one possible number of times than another.
In what follows therefore I shall take for granted that the rule given con-
cerning the event M in prop. 9. is also the rule to be used in relation to any
event concerning the probability of which nothing at all is known antecedently
to any trials made of observed concerning it. And such and event I shall call an
unknown event.
Cor. Hence, by supposing the ordinates in the ﬁgure AiB to be contracted
in the ratio of E to one. which makes no alteration in the proportion of the
parts of the ﬁgure intercepted between them, and applying what is said of the
event M to an unknown event, we have the following proposition, which gives
the rules of ﬁnding the probability of an event from the number of times it
actually happens and fails.
P R O P.
10.
If a ﬁgure be described upon any base AH (Vid. Fig.) having for it’s equation
y = xprq; where y, x, r are respectively the ratios of an ordinate of the ﬁgure
insisting on the base at right angles, of the segment of the base intercepted
between the ordinate and A the beginning of the base, and of the other segment
of the base lying between the ordinate and the point H, to the base as their
ommon consequent. I say then that if an unknown event has happened p times
and failed q in p + q trials, and in the base AH taking any two points as f
and t you erect the ordinates f c, tF at right angles with it, the chance that the
probability of an event lies somewhere between the ratio of Af to AH and that
of At to AH, is the ratio of tFCf , that part of the before-described ﬁgure which
is intercepted between the two ordinates, to ACFH the whole ﬁgure insisting on
the base AH.
This is evident from prop. 9. and the remarks made in the foregoing scholium
and corollary.
Now, in order to reduce the foregoing rule to practice, we must ﬁnd the
value of the area of the ﬁgure described and the several parts of it separated, by
ordinates perpendicular to its base. For which purpose, suppose AH = 1 and
HO the square upon AH likewise = 1, and Cf will be = y, and Af = x and
Hf = r, because y, x and r denote the ratios of Cf , Af , and Hf respectively to
∗It is here proved presently in art. 4. by computing in the method here mentioned that
AiB contracted in the ratio of E to 1 is to CA as 1 to n + 1 × E; from whence it plainly follows
that, antecedently to this contraction, AiB must be to CA in the ratio of 1 to n + 1, which is
a constant ratio when n is given, whatever p is.
12

AH. And by the equation of the curve y = xprq and (becauseAf + f H = AH)
p + 1
r + x = 1. Wherefore y = xp × 1 − x|q = xp − qx
q × q−1 × xp+2 −
+
2
q × q−1 × q − 13 × xp+3 + &c. Now the abscisse being x and the ordinate xp
2
the correspondent area is xp+1 (by prop. 10. cas. 1. Quadrat. Newt.)∗ and the
p+1
ordinate being qxp+1 the area is qxp+2 ; and in like manner of the rest. Wherefor,
p+2
the abscisse being x and the ordinate y or xp − qxp+1 + &c. the correspendent
area is xp+1 − q×xp+2 + q × q−1 × xp+3 − q × q−1 × q−2 × xp+2 + &c. Wherefore,
p+1
p+2
2
p+3
2
3
p+4
if x = Af = Af , and y = Cf = Cf , then ACf = ACf = xp+1 − q × xp+2 +
AH
AH
HO
p+1
p+2
q × q−1 × xp+3 − &c..
2
p+3
From which equation, if q be a small number, it is easy to ﬁnd the value of
the ratio of ACf to HO. and in like manner the value of the ratio of HCf to
HO is rq+1 − p × rq+2 + p × p−1 × rq+3 − p × p−1 × p−2 × rq+4 &c. which series
q+1
q+2
2
q+3
2
3
q+4
will consist of a few terms and therefor is to be used when p is small.
2. The same things supposed as before, the ratio of ACf to HO is xp+1 +
p+1
q× xp+2rq−1 + q× × q−1 × xp+3rq−2 + q × q−1 × q−2 × xp+4rq−3 + &c. +
p+1
p+2
p+1
p+2
p+3
p+1
p+2
p+3
p+4
xn+1 × q × q−1 × &c. × 1 where n = p + q. For this series is the same with
n+1
p+1
p+2
n
xp+1 − q × xp+2 &c. set down in Art. 1st as the value of the ratio of ACf to HO;
p+1
p+2
as will easily be seen by putting in the former instead of r its value 1 − x, and
expanding the terms and ordering them according to the powers of x. Or, more
reasily, by comparing the ﬂuxions of the two series, and in the former instead
of r substituting − ˙x∗.
3. In like manner, the ratio pf HCf to HO is rq+1xp + p × rq+2xp−1 + p ×
q+1
q+1
q+2
q+1
p−1 × rq+3xp−2 + &c.
q+2
q+3
4. If E be the coeﬃcient of that term of the binomial a + bp+q expanded in
which occurs at apbq, the ratio of the whole ﬁgure ACFH to HO is
1
× 1 , n
n+1
E
being = p + q. For, when Af = AH x = 1, r = 0. Wherefore, all the terms of
the series set down in Art. 2. as expressing the ratio of ACf to HO will vanish
except the last, and that becomes
1
× q × q−1 × &c. × 1 . But E being the
n+1
p+1
p+2
n
coeﬃcient in which occurs apbq is equal to p+1 × p+2 × &c. × n . And, because
q
q−1
1
Af is supposed to become = AH, ACf = ACH. From whence this article is
∗Tis very evident here, without having recourse to Sir Isaac Newton, that the ﬂuxion of the
area ACf being y ˙
x −qxp+1 ˙x+q × q−1 xp+2 ˙x&c., the ﬂuent or area itself is xp+1p+1−q×xp+2 ×
2
p+2
q × q−1 × xp+3 &c.
2
p+3
∗The ﬂuxion of the ﬁrst series is xprq ˙x+ qxp+1 p + 1+ qxp+1rq−1 ˙x +q × q−1 × xp+2rq−2 ˙r +
rp−1r
p+1
p+1
p+2
q
× q−1 × xp+2rq−2 ˙x + q × q−1 × q−3 × xp+3rq−3 ˙r &c. or, substuting − ˙x for r,
p+1
p+2
p+1
p+2
p+3
xprq ˙
x − qxp+1rq−1 ˙x + qxp+1rq−1 ˙x − q × q−1 × xp+2rq−2 ˙x + q × q−1 × xp+2rq−2 ˙x &c. which,
p+1
p+1
p+1
p+2
p+1
p+2
as all the terms after the ﬁrst destroy one another, is equal to xprq ˙
x = xp × 1 − x|q ˙x =
xp ˙
x × 1 − qx + q−1 x2 &c. = xp ˙xp+1 ˙x + q × q−1 xp+2 ˙x&c. = the ﬂuxion of the latter series
2
2
or of xp+1 − q × xp+2 &c. The two series therefore are the same.
p+1
p+2
13

plain.
5. The ratio of ACf to the whole ﬁgure ACFH is (by Art. 1. and 4.) n + 1 ×
E × xp+1 − q × xp+2 + +q × q−1 × xp+3 &c. and if, as x expresses the ratio of
p+1
p+2
2
p+3
Af to AH, X should express the ratio of At to AH; the ratio of AFt to ACFH
would be n + 1 × E × X − q Xp+2 + q × q−1 × Xp+3 − &c. and consequently
p+1
p+2
2
p+3
the ratio of tFCf to ACFH is n + 1 × E Xd into the diﬀerence between the two
series. Compare this with prop. 10. and we shall have the following practical
rule.
R U L E
1.
If noting is known concerning an event but that it has happened p times
and failed q in p + q or n trials, and from hence I guess the probability that
of its happening in a single time lies somewhere between any two degrees of
probability as X and x, the chance I am right in my guess is n + 1 × E Xd into
the diﬀerence between the series Xp+1 − q Xp+2 + q × Xp+2 + q × q−1 × Xp+3 − &c.
p+1
p+2
p+2
2
p+3
and the series xp+1 − q xp+2 + q × q−1 × xp+3 − &c. E being the coeﬃcient of
p+1
p+2
2
p+3
apbq when a + bn is expanded.
This is the proper rule to be used when q is a small number; but if q is large
and p small, change every where in the series here set down p into q and q into p
and x into r or 1 − x, and X into R = 1 − X; which will not make any alteration
in the diﬀerence between the two serieses.
Thus far Mr. Bayes’s essay.
With respect to the rule here given, it is further to be observed, that when
both p and q are very large numbers, it will not be possible to apply it in practice
on account of the multitude of terms which the serises in it will contain. Mr.
Bayes, therefore, by an investigation which it would be too tedious to give here,
has deduced from this rule another, which is as follows.
R U L E
2.
If nothing is known concerning an event but that it has happened p times
and failed q om p + q or n trials, and from hence I guess that the probability
of its happening in a single trial lies between p + z and p − z; if m2 = n3 ,
n
n
pq
a = p , b = q , E the coeﬃcient of the term which occurs at apbq when a + b|n
n
n
√
is expanded, and Σ = n+1 × 2pq
√
× E apbqXd by the series mz − m3z3 + n−2 ×
n
n
3
2n
m5z5 − n−2×n−4 × m7z7 + n−2 × n−2 × n−4 × n−6 × npz9 &c. my chance to be in
5
2n×3n
7
2n
2n
3n
4n
9
the right is greater than 1 + 2E apbq + 2 E apbq ∗ and less than
2 Σ
1−2 E apbq− 2 E apbq.
n
And if p = q my chance is 2 Σ exactly.
∗In Mr. Bayes’s manuscript this chance is made to be greater than
2 Σ
and less than
1+2 E apbq
2 Σ
. The third term in the two divisors, as I have given them, being omitted. But this
1−2 E apbq
being evidently owing to a small oversight in the deduction of this rule, which I have reason
to think Mr. Bayes had himself discovered, I have ventured to correct his copy, and to give
the rule as I am satisﬁed it ought to be given.
14

In order to render this rule ﬁt for use in all cases it is only necessary to know
how to ﬁnd within suﬃcient nearness the value of E apbq and also of the series
mz − M3z3 ∗. With respect to the former Mr.Bayes has proved that, supposing
3
K to signify the ratio of the quadrantal arc to it’s radius, E apbq will be equal
√
to
n
√
× by the ratio whose hyperbolic logarithm is 1 × 1 − 1 − 1 − 1 ×
2
Kpq
12
n
p
q
360
1 − 1 1 + 1 × 1 − 1 − 1 − 1 × 1 − 1 − 1 + 1 × 1 − 1 − 1
n3
p3 q3
1260
n5
p5
q5
1680
n7
p7
q7
1188
n9
p9
q9
&c. where the numeral coeﬃcients may be found in the following manner. Call
them A, B, C, D, E, &c. Then A =
1
=
1 . B =
1
− A. C =
2. 2. 3
3. 4
2. 4. 5
3
1
− 10B+A. D = 1 − 35C+21B+A. E =
1
− 126C+84D+36B+A. F =
2. 6. 7
5
2. 8. 9
7
2. 10. 11
9
1
− 462D+330C+165E+55B+A &c. where the coeﬃcients of B, C, D, E, F, &c.
2. 12. 13
11
in the values of D, E, F, &c. are the 2, 3, 4, &c. highest coeﬃcients in a + b|7,
a + b|9, a + b|11, &c. expanded; aﬃxing in every particular value the least of
these coeﬃcients to B, the next in magnitude to the furthest letter from B, the
next to C, the next to the furthest but one, the next to D, the next to the
furthest but two, and so on†.
With respect to the value of the series mz − m3z3 + n−2 × m5z5 &c. he has
3
2n
5
observed that it may be calculated directly when mz os less htan 1, or even not
√
greater than
3: but when mz is much larger it becomes impracticable to do
this; in which case he shews a way of easily ﬁnding two values of it very nearly
equal between which it’s true value must lie.
The theorem he gives for this purpose is as follows.
Let K, as before, stand for the ratio of the quadrant arc to its radius, and H
for the ratio whose hyperbolic logarithm is 22−1 − 24−1 + 26−1 − 28−1 &c.
2n
360n3
1260n5
1680n7√
Then the series mz − m3z3 &c. will be greater or less than the series Hn × K
√ −
3
n+1
2
n +1
n +2
n +3
n
× 1−2m2z2 | 2
| 2
| 2
n
+ n2 × 1− 2m2z2
n
+ 3n3 × 1− 2m2z2
n
+ 2×5×n4 ×
n+2
2mz
n+2
n+4×4m3z3
n+2
n+4×n+6×8m5z7
n+2
1− 2m2z2 | n +4
2
n
− &c. continued to any number of terms, according as the
n+4×n+6×n+8×16m7z7
last term has a positive or a negative sign before it.
From substuting these values of E apbq and mz − m3z3 + + n−2 × m5z5 &c.
3
2n
5
in the 2d rule arises a 3d rule, which is the rule to be used when mz is of some
considerable magnitude.
R U L E
3.
∗A very few terms of this series will generally give the hyperbolic logarithm to a suﬃcient
degree of exactness. A similar series has been given by Mr. De Moivre, Mr. Simpson and other
eminent mathematicians in an expression for the sum of the logarithms of the numbers 1, 2, 3,
4, 5, to x, which sum they have asserted to be equalto 1 log .c + x + x + 1 × log .x − x + 1 −
2
2
12x
1
+
1
&c. c denoting the circumference of a circle whose radius is unity. But Mr.
360x3
1260x5
Bayes, in a preceding paper in this volume, has demonstrated that, though this expression will
very nearly approach to the value of this sum when only a proper number of the ﬁrst terms is
taken, the whole series cannot express any quantity at all, beause, let x be what it will, there
will always be a part of the series where it will begin to diverge. This observation, though it
does not much aﬀect the use of this series, seems well worth the noticeof mathematicians.
†This method of ﬁnding thesecoeﬃcients I have deduced from the demonstration of the
third lemma at the end of Mr. Simpson’s Treatise on the Nature and Laws of Chance.
15

If nothing is known of an event but that it has happened p times and failed
q in p + q or n trials, and from hence I judge that the probability of it’s hap-
pening in a single trial lies between p + z and p − z my chance to be right
n
n
√
√
is greater than
Kpq×h
2
√
×2H − √ × n+1 × 1 × 1 − 2m2z2 |n+1
2
and
n+2
mz
n
2
Kpq+hn 1 +hn− 1
2
K
2
√
√
less than
Kpq×h
2
√
multiplied by the 3 terms 2H − √ × n+1 × 1 ×
n+2
mz
2
Kpq−hn 1 −hn− 12
K
2
√
1 − 2m2z2 | n +1
2
+2
2
+ √
× n × n+1 × 1 × 1 − 2m2z2 |n2
where m2, K, h and
n
K
n+2
n+4
m3z3
n
H stand for the quantities already explained.
An A P P E N D I X.
CONTAINING
An Application of the foregoing Rules to some particular Cases
THE ﬁrst rule gives a direct and perfect solution in all cases; and the two
following rules are only particular methods of approximating to the solution
given in the ﬁrst rule, when the labour of applying it becomes too great.
The ﬁrst rule may be used injall cases where either p or q are nothing or
not too large. The second rule may be used in all cases where mz is less than
√3; and the 3d in all cases where m2z2 is greater than 1 and less than n, if n
2
is an even number and very large. If n is not too large this last rule cannot be
much wanted, because, m decreasing continually as n is diminished, the value
of z may in this case be taken large, (and therefore a considerable interval had
between p − z and p + z) and yet the operation be carried on by the 2d rule;
n
n
√
or mz not exceed
3.
But in order to shew distinctly and fully the nature of the present problem,
and how far Mr. Bayes has carried the solution of it; I shall give the result of
this solution in a few cases, beginning with the lowest and most simple.
Let us then ﬁrst suppose, of such and event as that called M in the essay,
or an event about the probability of which, antecedently to trials, we know
nothing, that it has happened once, and that it is enquired what conclusion
we may draw from hence with respct to the probability of it’s happening on a
second trial.
The answer is that there would be an odds of three to one for somewhat
more than an even chance that it would happen on a second trial.
For in this case, and in all others where q is nothing, the expression n + 1 ×
Xp+1 − xp+1 or Xp+1 − xp+1 gives the solution, as will appear from considering
p+2
p+1
the ﬁrst rule. Put therefore in this expression p + 1 = 2, X = 1. and x = 1 and
2
it will be 1 − 1|2 or 1 ; which shews the chance there is that the probability of
4
an event that has happened once lies somewhere between 1 and 1 ; or (which is
2
the same) the odds that it is somewhat more than an even chance that it will
16

happen on a second trial∗. In the same manner it will appear that if the event
has happened twice, the odds now mentioned will be seven to one; if thrice,
ﬁfteen to one; and in general, if the event has happened p times, there will be
an odds of 2p+1 − 1 to one, for more than an equal chance that it will happen
on further trials.
Again, suppose all I know of an event to be that it has happened ten times
without failing, and the enquiry to be what reason we shall have to think we
are right if we guess that the probability of it’s happening in a single trial lies
somewhere between 16 and 2 , or that the ratio of the causes of it’s happening
17
3
to those of it’s failure is some ratio between that of sixteen to one and two to
one.
Here p + 1 = 11, X = 16 and x = 2 and Xp+1 − xp+1 = 16 |11 − 2 |11 = .5013
17
3
17
3
&c. The answer therefore is, that we shall have very nearly an equal chance for
being right.
In this manner we may determine in any case what conclusion we ought to
draw from a given number of experiments which are unopposed by contrary
experiments. Every one sees in general that there is reason to expect an event
with more or less conﬁdence according to the greater of less number of times in
which, under given circumstances, it has happened without failing; but we here
see exactly what this reason is, on what principles it is is founded, and how we
ought to regulate our expectations.
But it will be proper to dwell longer on this head.
Suppose a solid or die or whose number of sides and constitution we know
nothing; and that we are to judge of these from experiments made in throwing
it.
In this case, it should be observed, that it would be in the highest degree
improbable that the solid should, in the ﬁrst trial, turn any one side which
could be assigned before hand; because it would be known that some side must
turn, and that there was an inﬁnity of sides, or sides otherwise marked, which
it was equally likely that it should turn. The ﬁrst throw only shews that it
has the side then thrown, without giving any reason to think that it has any
number of times rather than any other. It will appear, therefore, that after
the ﬁrst throw and not before, we should be in the circumstances required by
the conditions of the present problem, and that the whole eﬀect of this throw
would be to bring us into these circumstances. That is: the turning the side ﬁrst
thrown in any subsequent single trial would be an event about the probability or
improbability of which e could form no judgment, and of which we should know
no more than that it lay somewhere between nothing and certainty. With the
second trial then our calculations must begin; and if in that trial the supposed
solid turns again the same side, there will arise the probability of three to one
that it has more of that sort of sides than of all others; or (which comes to the
same) that there is somewhat in its constitution disposing it to turn that side
oftenest: And this probability will increase, in the manner already explained,
∗There can, I suppose, be no reason for observing that on this subject unity is always made
to stand for certainty, and 1 for an even chance.
2
17

with the number of times in which that side has been thrown without failing
It should not, however, be imagined that any number of such experiments can
give suﬃcient reason for thinking that it would never turn any other side. For,
suppose it has turned the same side in every trial a million of times. In these
circumstances there would be an improbability that it had less than 1.400,000
more of these sides than all others; but there would also be an improbability
that it had above 1.600,000 times more. The chance for the latter is expressed
by 1600000 raised to the millioneth power subtracted from unity, which is equal
1600001
to .4647 &c. and the chance for the former is equal to 1400000 raised to the
1400001
same power, or to .4895; which, being both less than an equal chance, proves
what I have said. But thouth it would be thus improbable that it had above
1.600,000 times more or less than 1.400,000 times more of these sides than of
all others, it by no means follows that we have any reason for judging that the
true proportion in this case lies somewhere between that of 1.600,000 to one and
1.400,000 to one. For he that will take the pains to make the calculation will
ﬁnd that there is nearly the probability experessed by .527, or but little more
than an equal chance, that it lies somewhere between that ot 600,000 to one and
three millions to one. It may deserve to be added, that it is more probable that
this proportion lies somewhere between that of 900,000 to 1 and 1.900,000 to 1
than between any other two proportions whose antecedents are to one another
as 900,000 to 1.900,000, and consequents unity.
I have made these observattions chieﬂy because they are all strictly applica-
ble to the events and appearances of nature. Antecedently to all experience, it
would be improbable as inﬁnite to one, that any particular event, before-hand
imagined, should follow the application of any one natural object to another;
because there would be an equal chance for any one of an inﬁnity of other events.
But it we had once seen any particular eﬀects, as the burning of wood on putting
ity into ﬁre, or the falling of a stone on detaching it from all contiguous objects,
then the conclusions to be drawn from any number of subsequent events of the
same kind would be determined in the same manner with the conclusions just
mentioned relating to the constitution of the solid I have supposed.——In other
words. The ﬁrst experiment supposed to be ever made on any natural object
would only inform us of one event that may follow a particular chance in the
circumstances of those objects; but it would not suggest to us any ideas of uni-
formity in nature, or give use the least reason to apprehend that it was, in that
instance or in any other, regular rather than irregular in its operations. But it
the same event has followed without interruption in any one or more subsequen-
t experiments, then some degree of uniformity will be observed; reason will be
given to expect the same success in further experiments, and the calculations
directed by the solution of this problem may be made.
One example here it will not be amiss to give.
Let us imagine to ourselves the case of a person just brought forth into this,
world and left to collect from his observations the order and course of events
what powers and causes take place in it. The Sun would, probably, be the ﬁrst
object that would engage his attention; but after losing it the ﬁrst night he would
be entirely ignorant whether he should ever see it again. He would therefore be
18

in the condition of a person making a ﬁrst experiment about an event entirely
unknown to him. But let him see a second appearance or one return of the Sun,
and an expectation would be raised in him of a second return, and he might
know that there was an odds of 3 to 1 for some probability of this. This odds
would increase, as before represented, with the number of returns to which he
was witness. But no ﬁnite number of returns would be suﬃcient to produce
absolute or physical certaintly. For let it be supposed that he has seen it return
at regular and stated intervals a million of times. The conclusions this would
warrant would be such as follow—— There would be the odds of the millioneth
power of 2, to one, that it was likely that it would return again at the end of
the usual interval. There would be the probability expressed by .5352, that the
odds for this was not greater than 1.600,000 to 1; And the probability expressed
by .5105, that it was not less than 1.400,000 to 1.
It should be carefully remembered that these deductions suppose a previous
total ignorance of nature. After having observed for some time the course of
events it would be found for some time the course of events it would be found
that the operations of nature are in general regular, and that the powers and
laws which prevali in it are stable and parmanent. The consideration of this will
cause one or a few experiments often to produce a much stronger expectation
of success in further experiments than would otherwise have been reasonable;
just as the frequent observation that things of a sort are disopposed together
in any place would lead us to conclude, upon discovering there any object of a
particular sort, that there are laid up with it many others of the same sort. It
is obvious that this, so far from contradicting the foregoing deductions, is only
one particular case to which they are to be applied.
What has been said seems suﬃcient to shew us what conclusions to draw
from uniform experience. It demonstrates, particularly, that instead of proving
that events will always happen agreeably to it, there will be always reason
against this conclusion. In other words, where the course of nature has been the
most constant, we can have only reason to reckon upon a recusrrency of events
proportioned to the degree of this constancy, but we can have no reason for
thin king that there are no causes in nature which will ever interfere with the
operations the causes from which this constancy is derived, or no circumstancce
of thw world in which it will fail. And if this is true, supposing our only data
derived from experience, we shall ﬁnd additional reason for thinking thus if
we apply other principles, or have recourse to such considerations as reason,
independently of experience, can suggest.
But I have gone further than I intended here; and it is time to turn our
thoughts to another branch of this subject: I mean, to cases where an experiment
has sometimes succeeed and sometimes failed.
Here, again, in order to be as plain and explicit as possible, it will be proper
to put the following case, which is the easiest and simplest I can think of.
Let us then imagine a person present at the drawing of a lottery, who knows
nothing of its scheme or of the proportion of Blanks to Prizes in it. Let it
further be supposed, that he is obliged to infer this from the number of blanks
he hears drawn compared with the number of prizes; and that it is enquired
19

what conclusions in these circumstances he may reasonably make.
Let him ﬁrst hear ten blanks drawn and one prize, and let it be enquired
what chance he will have for being right if he gussses that the proportion of
blanks to prizes in the lottery lies somewhere between the proportions of 9 to 1
and 11 to 1.
Here taking X = 11 , x = 9 , p = 10, q = 1, n = 11, E = 11, the required
12
10
chance, according to the ﬁrst rule, is n + 1 × E into the diﬀerences between
Xp+1 − qXp+2
|11
|11
and xp+1 − qxp+2 = 12 × 11 × 11
12
− 11 |1212− 910
− 9 |1212 =
p+1
p+2
p+1
p+2
11
12
11
10
.07699 &c. There would therefore be an odds of about 923 to 76 against his
being right. Had he guessed only in general there were less than 9 blanks to a
prize, there would have been a probability of his being right equal to .6589, or
the odds of 65 to 34.
Again. suppose that he has heard 20 blanks drawn and 2 prizes; what chance
will he have for being right if he makes the same guess?
Here X and x being the same, we have n = 22, p = 20, q = 2, E = 231,
and the required chance equal to n + 1 × E × Xp+1 − q xp+2+ q × q−1 × Xp+3 −
p+1
p+2
2
p+3
xp+1 − qxp+2 + q × q−1 × xp+3 = .10843 &c.
p+1
p+2
2
p+3
He will, therefore, have a better chance for being right in the former instance,
the oddes against him now being 892 to 108 or about 9 to 1. But should b=he
only guess in general, as before, that there were less than 9 blanks to a prize,
his chance for being right will be worse; for instead of .6589 or an odds of near
two to one, it will be .584, or an odds of 584 to 415.
Suppose, further, that he has heard 40 blanks drawn and 4 prizes; what will
the before-mentioned chances be?
The answer here is .1525, for the former of these chances; and .527, for
the latter. There will, therefore, now be an odds of only 5 1 to 1 against the
2
proportion of blanks to prizes lying between 9 to 1 and 11 to 1; and but little
more than an equal chance that it is less than 9 to 1.
Once more. Suppose he has heard 100 blanks drawn and 10 prizes.
The answer here may still be found by the ﬁrst rule; and the chance for a
proportion of blanks to prizes less than 9 to 1 will be .44109, and for a proportion
graeter than 11 to 1 .3082. It would therefor be likely that there were not fewer
than 9 or more than 11 blanks to a prize. But at the same time it will remain
unlikely ∗ that the true proportion should lie between 9 to 1 and 11 to 1, the
chance for this being .2506 &c. There will therefore be still an odds of near 3
to 1 against this.
From these calculations it appears that, in the circumstances I have sup-
posed, the chance for being right in guessing the proportion of blanks to prizes
to be nearly the same with that of the number of blanks drawn in a given time to
∗I suppose no attentive person will ﬁnd any diﬃculty in this. It is only saying that,
supposing the intervalbetween nothing and certainty divided into a hundred equalchances,
there will be 44 of them for a less proportion of blanks to prizes than 9 to 1, 31 for a greater
than 11 to 1; in which it is obvious that, though though one of these suppositions must be true,
yet, having each of them more chances against them than more them, they are all separately
unlikely.
20

the number of prizes drawn, is continually increasing as these numbers increase;
and therefore, when they are considerably large, this conclusion may be lloked
upon as moreally certain. By parity of reason, it follows universally, with respect
to every element about which a great number of experiments has been made,
that the causes of its happening bear the same proportion to the causes of its
failing, with the number of happenings to the number of failures; and that. if an
event whose casses are supposed to be known, happens oftener or seldomer to be
known, happens oftener or seldomer than is agreeable to this conclusion, there
will be reason to believe that there are some unknown causes which disturb
the operations of the known ones. With respect, therefore, particularly to the
course of events in nature, it appears, that there is demonstrative evidence to
prove that order of events which we observe, and not from any of the powers of
chance∗. This is just as evident as it would be, in the case I have insisted on.
that the reason of drawing 10 times more blanks than prizes in millions of trials,
was, that there were in the wheel about so many more blanks than prizes.
But to proceed a little further in the demonstration of this point.
We have seen that supposing a person, ignorant of the whole scheme of
a lottery, should be led to conjecture, from hearing 100 blanks and 10 prizes
drawn, that the proportion of blanks to prizes in the lottery was somewhere
between 9 to 1 and 11 to 1, the chance for his being right would be .2506 &c.
Let now enquire what this chance would be in some higher cases.
Let it be supposed that blanks have been drawn 1000 times, and prizes 100
times in 1100 trials.
In this case the powers of X and x rise to high, and the number of terms in
the two serieses Xp+1 − qXp+1 &c. and xp+1 − qxp+2 &c. become so numerous
p+1
p+2
p+1
p+2
that it would require immense labout to obtain the answer by the ﬁrst rule.
’Tis necessary, therefore, to have recourse to the second rule. But in order to
make use of it, the interval between X and x must be a little aleterd. 10 − 9
11
10
is 1 , and therefore the interval between 10 − 1 and 10 + 1 will nearly be
110
1
110
11
110
the same with the interval between 9 and 11 , only somewhat larger. If then
10
12
we make the question to be; what chance there would be (supposing no more
known than that blanks have been drawn 1000 times and prizes 100 times in
1100 trials) that the probability of drawing a blank in a single trial would lie
somewhere between 10 − 1 and 10 + 1 we shall have a question of the same
11
110
11
110
kind with the previous questions, and deviate but little from the limits assigned
in them.
The answer, according to the second rule, is that this chance is greater than
√
2 Σ
n and less than
2 Σ
n, E being n+1 ×
2pq
√
×
1−2 E apbq+ 2 E apbq
1−2 E apbq− 2 E apbq
n
n
E appq × mz − m3z3 + n−2 × m5z5 &c.
3
2n
5
√
By making here 1000 = p 100 = q 1100 = n
1
= z, m =
n3
√
=
110
pq
√
1.048808, E apbq = b ×
n , b being the ratio whose hyperbolic logarithm is
2
Kpq
1 × 1 − 1 − 1 − 1 × 1 − 1 − 1 + 1 × 1 − 1 − 1 &c. and K the ratio
12
n
p
q
360
n3
p3
q3
1260
n5
p5
q5
∗See Mr. De Moivre’s Doctrine of Chances, pag. 250.
21

of the quadrantal to radius; the former of these expressions will be found to be
.7953, and the latter .9405 &c. The chance enquired after, therefore, is greater
than .7953, and less than .9405. That is; there will be an odds for being right
in guessing that the proportion of blanks to prizes lies nearly between 9 to 1
and 11 to 1, (or exactly between 9 to 1 and 1111 to 99) which is greater than 4
to 1, and less than 16 to 1.
Suppose, again, that no more is known than that blanks have been drawn
10,000 times and prizes 1000 times in 11000 trials; what will the chance now
mentioned be?
Here the second as well as the ﬁrst rule becomes useless, the value of mz
being so great as to render it scarcely possible to calculate directly the series
mz−m3z3 + n−2 × m5z − &c. The third rule, therefore, must be used; and the
3
2n
5
information it gives us is, that the required chance is greater than .97421, or
more than an odds of 40 to 1.
By calculations similar to these may be determined universally, what expec-
tations are warranted by any experiments, according to the diﬀerent number
of times in which they have succeeded and failed; or what should be thought
of the probability that any particular cause in nature, with which we have any
acquaintance, will or will not, in any single trial, produce an eﬀect that has
been conjoined with it.
Most persons, probably, might expect that the chances in the specimen I have
given would have been greater than I have found them. But this only shews
how liable we are to be in error when we judge on this subject independently
of calculation. One thing, however, should be remembered here; and that is,
the narrowness of the interval between 10 + 1 and 10 − 1 . Had this interval
11
110
11
110
been taken a little larger, there would have been a considerable diﬀerence in the
results of the calculations. Thus had it been taken double, or z = 1 , it would
55
have been found in the fourth instance that instead of odds against that instead
of odds against there were odds for being right in judging that the probability
of drawing a blank in a single trial lies between 10 + 1 and 10 − 1 .
11
55
11
55
The foregoing calculations further shew us the uses and defects of the rules
laid down in the essay. ’Tis evident that the two last rules do not give us the
required chances within such narrow limits as could be wished. But here again
it should be considered, that these limits become narrower and narrower as q
is taken larger in respctof p; and when p and q are equal, the exact solution is
given in all cases by the second rule. These two rules therefore aﬀord a direction
to our judgment that may be of considerable use till some person shall discover
a better approximation to the value of the two series’s in the ﬁrst rule†.
But what most of all recommends the solution in this Essay is, that it is
compleat in those cases where information is most wanted, and where Mr. De
Moivre’s solution of the inverse problem can give little of no direction; I mean,
†Since this was written I have found a method of considerably improving the approximation
in the 2d and 3d rules by demonstrating the expression
2 Σ
comes almost as
1+2 E apbq + 2 E apbq
n
near to the true value wanted as there is reason to desire, only always somewhat less. It seems
necessary to hint this here; though the proof of it cannot be given.
22

in all cases where either p or q are of no considerable magnitude. In other cases,
or when p and q are very considerable, it is not diﬃcult to perceive the truthe of
what has been here demonstrated, or that there is reason to believe in general
that the chances for the happening of an event are to the chances for its failure
in the same ratio with that of p to q. But we shall be greatly deceived if we
judge in this manner when either p or q are small. And tho’ in such cases the
Data are not suﬃcient to discover the exact probability of an event. yet it is
very agreeable to be able to ﬁnd the limits between which it is reasonable to
think it must lie, and also to be able to determine the precise degree of assent
which is due to any conclusions or assertions relating to them.
[Philosophical Transactions of the Royal Society of London 53 (1763), 370–418.]
23