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 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 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 ar 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 ircumstances, it has happened a certain number of times, and failed a certai 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 or which he had argued might not perhaps be looked upon by all as reasonable nd 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 o 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 experiments we have to support 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
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 Society, 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 probability, 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 ecessary 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 ignorant; so it is, in all probability, the ource 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 time 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 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. Ir. 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 he 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 ery case of 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. 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. tSee his Doctrine of Chances, p. 252, &e
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 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 ssay 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 carrie <pect that you should minutely examine every part of what I now send o/y at your time is so muc ch taken up that i cannot reasona 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 Ir. 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 everycase 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
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
PROBLEM 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 lie somewhere between any two degrees of probability that can be named SECTION 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 happer 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 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 do neither increase nor abate the probability of the rest PROP 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 receive N, and that the probability of the lst, 2d, and 3d are respectively N,N,N. Then(by definition of probability) the value of my expectation from the lst will be a. from the 2d 6. and from the 3d c. herefore 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 4*+c or N+N+N. The sum of the probabilities of each of them 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 an event is f that of it's failure will be N-ps. Wherefore if the PROP 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
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 happens, 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 N , b N , c N . Then (by definition of probability) the value of my expectation from 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 N or a N + b N + c N . The sum of the probabilities of each of them. 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 N that of it’s failure will be N−P 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 N. Then(by definition 5)the value of his expectation 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 N, that of its failure(by corollary prop. 1)is >N. But An is ton as P is to N-P, i.e. the probability of the event is to the probability of it's failure, as his loss if it fails to his gain if it happens PROP 3 The probability that two subsequent events will both happen is a ratio com- pounded of the probability of the Ist, and the probability of the 2d on suppo- sition the lst happens Suppose that, if both events happen, I am to receive n, that the probabilit both will happen is N, that the 1st will is n (and consequently that the 1st will not is -N) and that the 2d will happen upon supposition the lst does is Then(by definition 5)P will be the value of my expectation, which will become b is the lst happens. Consequently if the lst happens, my gain is b-P, and if it fails my loss is P. Wherefore, by the foregoing proposition, n is to n, i.e. 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 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 lst and the probability of N, and the probability of both together be n, then the probability of the 2d on supposition the lst happens is PROP 4 If there be two subesequent events be determined every day, and each day the probability of the 2d is s and the probability of both f, and I am to receive N if both of the events happen the lst day on which the 2d does: I say, according to these conditions, the probability of my obtaining n is f. For if not, let the probability of my obtaining n bef and let y be to r as n-b to N. The since x is the probability of my obtaining n(by definition 1)a is the value of my expectation. And again, because according to the foregoing conditions the lst day I have an expectation of obtaining n depdening on the happening of both events together, the probability of which is f, the value of this expectation is 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 a depending on the failure of the 2d event the probability of which(by cor prop 1)is NN or 3, because y is to r as N-b to N. Wherefore since r is the thing expected and the probability of obtaining it, the value of this expectation is
Suppose a person has an expectation of receiving N, depending on an event the probability of which is P N . Then (by definition 5) the value of his expectation 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 N , that of its failure (by corollary prop. 1) is N−P N . But N−P N is to P N as P is to N − P, i.e. the probability of the 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 compounded of the probability of the 1st, and the probability of the 2d on supposition the 1st happens. Suppose that, if both events happen, I am to receive N, that the probability both will happen is P N , that the 1st will is a N (and consequently that the 1st will not is N−a N ) and that the 2d will happen upon supposition the 1st does is b 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 N is to N−a N , i.e. 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 N , and the probability of both together be P N , then the probability of the 2d on 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 N and the probability of both P N , and I am to receive 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 b . For if not, let the probability of my obtaining N be x N and let y be to x as N − b to N. The since x N is the probability of my obtaining N (by definition 1) x is the value of my 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 N , the value of this expectation is 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 N or y x, because y is to x as N − b to N. Wherefore since x is the thing expected and y x the probability of obtaining it, the value of this expectation is 5