therefore assume the probability of each event as 1/2. But if it shows head in the first throw, we now have very slight experimental evidence in favour of a tendency to show head. The chance of two heads is now slightly greater than 1/4, which it appeared to be at first,23 and as we go on throwing the coin time after time, the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experience. As Laplace remarks, we ought always to have regard to such considerations in common life. Events when closely scrutinized will hardly ever prove to be quite independent, and the slightest preponderance one way or the other is some evidence of connection, and in the absence of better evidence should be taken into account.
The grand object of seeking to estimate the probability of future events from past experience, seems to have been entertained by James Bernoulli and De Moivre, at least such was the opinion of Condorcet; and Bernoulli may be said to have solved one case of the problem.24 The English writers Bayes and Price are, however, undoubtedly the first who put forward any distinct rules on the subject.25 Condorcet and several other eminent mathematicians advanced the mathematical theory of the subject; but it was reserved to the immortal Laplace to bring to the subject the full power of his genius, and carry the solution of the problem almost to perfection. It is instructive to observe that a theory which arose from petty games of chance, the rules and the very names of which are forgotten, gradually advanced, until it embraced the most sublime problems of science, and finally undertook to measure the value and certainty of all our inductions.
Fortuitous Coincidences.
We should have studied the theory of probability to very little purpose, if we thought that it would furnish us with an infallible guide. The theory itself points out the approximate certainty, that we shall sometimes be deceived by extraordinary fortuitous coincidences. There is no run of luck so extreme that it may not happen, and it may happen to us, or in our time, as well as to other persons or in other times. We may be forced by correct calculation to refer such coincidences to a necessary cause, and yet we may be deceived. All that the calculus of probability pretends to give, is the result in the long run, as it is called, and this really means in an infinity of cases. During any finite experience, however long, chances may be against us. Nevertheless the theory is the best guide we can have. If we always think and act according to its well-interpreted indications, we shall have the best chance of escaping error; and if all persons, throughout all time to come, obey the theory in like manner, they will undoubtedly thereby reap the greatest advantage.
No rule can be given for discriminating between coincidences which are casual and those which are the effects of law. By a fortuitous or casual coincidence, we mean an agreement between events, which nevertheless arise from wholly independent and different causes or conditions, and which will not always so agree. It is a fortuitous coincidence, if a penny thrown up repeatedly in various ways always falls on the same side; but it would not be fortuitous if there were any similarity in the motions of the hand, and the height of the throw, so as to cause or tend to cause a uniform result. Now among the infinitely numerous events, objects, or relations in the universe, it is quite likely that we shall occasionally notice casual coincidences. There are seven intervals in the octave, and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals. It is hardly yet decided whether this apparent coincidence, with which Newton was much struck, is well founded or not,26 but the question will probably be decided in the negative.
It is certainly a casual coincidence which the ancients noticed between the seven vowels, the seven strings of the lyre, the seven Pleiades, and the seven chiefs at Thebes.27 The accidents connected with the number seven