If an event has happened *m* times and failed *n* times, the probability that it will happen on the next occasion is (*m* + 1)/(*m* + *n* + 2). Thus, if we assume that of the elements discovered up to the year 1873, 50 are metallic and 14 non-metallic, then the probability that the next element discovered will be metallic is 51/66. Again, since of 37 metals which have been sufficiently examined only four, namely, sodium, potassium, lanthanum, and lithium, are of less density than water, the probability that the next metal examined or discovered will be less dense than water is (4 + 1)/(37 + 2) or 5/39. We may state the results of the method in a more general manner thus,[169]--If under given circumstances certain events A, B, C, &c., have happened respectively *m*, *n*, *p*, &c., times, and one or other of these events must happen, then the probabilities of these events are proportional to *m* + 1, *n* + 1, *p* + 1, &c., so that the probability of A will be (*m* + 1)/(*m* + 1 + *n* + 1 + *p* + 1 + &c.) But if new events may happen in addition to those which have been observed, we must assign unity for the probability of such new event. The odds then become 1 for a new event, *m* + 1 for A, *n* + 1 for B, and so on, and the absolute probability of A is (*m* + 1)/(1 + *m* + 1 + *n* + 1 + &c.) [169] De Morgan’s *Essay on Probabilities*, Cabinet Cyclopædia, p. 67. It is interesting to trace out the variations of probability according to these rules. The first time a casual event happens it is 2 to 1 that it will happen again; if it does happen it is 3 to 1 that it will happen a third time; and on successive occasions of the like kind the odds become 4, 5, 6, &c., to 1. The odds of course will be discriminated from the probabilities which are successively 2/3, 3/4, 4/5, &c. Thus on the first occasion on which a person sees a shark, and notices that it is accompanied by a little pilot fish, the odds are 2 to 1, or the probability 2/3, that the next shark will be so accompanied.