Probabilities may be added to or subtracted from each other under the important condition that the events in question are exclusive of each other, so that not more than one of them can happen. It might be argued that, since the probability of throwing head at the first trial is 1/2, and at the second trial also 1/2, the probability of throwing it in the first two throws is 1/2 + 1/2, or certainty. Not only is this result evidently absurd, but a repetition of the process would lead us to a probability of 1-1/2 or of any greater number, results which could have no meaning whatever. The probability we wish to calculate is that of one head in two throws, but in our addition we have included the case in which two heads appear. The true result is 1/2 + 1/2 × 1/2 or 3/4, or the probability of head at the first throw, added to the exclusive probability that if it does not come at the first, it will come at the second. The greatest difficulties of the theory arise from the confusion of exclusive and unexclusive alternatives. I may remind the reader that the possibility of unexclusive alternatives was a point previously discussed (p. 68), and to the reasons then given for considering alternation as logically unexclusive, may be added the existence of these difficulties in the theory of probability. The erroneous result explained above really arose from overlooking the fact that the expression “head first throw or head second throw” might include the case of head at both throws. *The Logical Alphabet in questions of Probability.* When the probabilities of certain simple events are given, and it is required to deduce the probabilities of compound events, the Logical Alphabet may give assistance, provided that there are no special logical conditions so that all the combinations are possible. Thus, if there be three events, A, B, C, of which the probabilities are, α, β, γ, then the negatives of those events, expressing the absence of the events, will have the probabilities 1 - α, 1 - β, 1 - γ. We have only to insert these values for the letters of the combinations and multiply, and we obtain the probability of each combination. Thus the probability of ABC is αβγ; of A*bc*, α(1 - β)(1 - γ).