One combination gives 0 tail. Probability 1/8. Three combinations gives 1 tail. Probability 3/8. Three combinations give 2 tails. Probability 3/8. One combination gives 3 tails. Probability 1/8. We can apply the same considerations to the imaginary causes of the difference of stature, the combinations of which were shown in p. 188. There are altogether 128 ways in which seven causes can be present or absent. Now, twenty-one of these combinations give an addition of two inches, so that the probability of a person under the circumstances being five feet two inches is 21/128. The probability of five feet three inches is 35/128; of five feet one inch 7/128; of five feet 1/128, and so on. Thus the eighth line of the Arithmetical Triangle gives all the probabilities arising out of the combinations of seven causes. *Rules for the Calculation of Probabilities.* I will now explain as simply as possible the rules for calculating probabilities. The principal rule is as follows:-- Calculate the number of events which may happen independently of each other, and which, as far as is known, are equally probable. Make this number the denominator of a fraction, and take for the numerator the number of such events as imply or constitute the happening of the event, whose probability is required. Thus, if the letters of the word *Roma* be thrown down casually in a row, what is the probability that they will form a significant Latin word? The possible arrangements of four letters are 4 × 3 × 2 × 1, or 24 in number (p. 178), and if all the arrangements be examined, seven of these will be found to have meaning, namely *Roma*, *ramo*, *oram*, *mora*, *maro*, *armo*, and *amor*. Hence the probability of a significant result is 7/24. We must distinguish comparative from absolute probabilities. In drawing a card casually from a pack, there is no reason to expect any one card more than any other. Now, there are four kings and four queens in a pack, so that there are just as many ways of drawing one as the other, and the probabilities are equal. But there are thirteen diamonds, so that the probability of a king is to that of a diamond as four to thirteen. Thus the probabilities of each are proportional to their respective numbers of ways of happening. Again, I can draw a king in four ways, and not draw one in forty-eight, so that the probabilities are in this proportion, or, as is commonly said, the *odds* against drawing a king are forty-eight to four. The odds are seven to seventeen in favour, or seventeen to seven against the letters R,o,m,a, accidentally forming a significant word. The odds are five to three against two tails appearing in three throws of a penny. Conversely, when the odds of an event are given, and the probability is required, *take the odds in favour of the event for numerator, and the sum of the odds for denominator*.