THE THEORY OF PROBABILITY 179 7. Two balls are drawn at the same time from a bag containing 3 white and 5 black balls. What is the probability that both will be white ? that both will be black ? that one will be black and one white ? Suggestion. There are C(^) possible pairs of balls and CQ) possible pairs of white balls. 8. What is the chance of throwing one, and only one, 5 with one throw of two dice ? 66. Total and partial probability. The probability that a given event in a series of n mutually exclusive events will happen and all the others fail is called partial, or relative, probability, and the probability that any event whatever of the series will happen and the others fail is called total probability. To illustrate, suppose a bag contains 11 white balls, of which 5 are marked with crosses and 6 not; 7 black balls, 4 with and 3 without crosses; 7 yellow balls, 2 with and 5 without crosses. The drawing of a ball with a cross occurs with any one of a series of three independent events: viz., (1) drawing a white ball with a cross, (2) drawing a black ball with a cross, (3) drawing a yel- low ball with a cross. The probability that a given one of these three events — for example, drawing a white ball with a cross — will happen is partial probability, while the probability of draw- ing a ball with a cross" regardless of color is total probability. In the foregoing examples there are four things that interest us: 1. The probability of drawing a white ball with a cross is A. 2. The probability of drawing a black ball with a cross is A. 3. The probability of drawing a yellow ball with a cross is -fr. 4. The probability of drawing a ball with a cross is ^. The striking fact is that the sum of the partial probabilities, ^, ^, and ^, is equal to the total probability. This fact is true in the general case and forms one of the two fundamental rules for the computation of probabilities. THEOREM. The total probability of an event is equal to the sum of its partial probabilities. Proof. Suppose that there are m possible cases and n events in the series. Let a^, a^, ; (»„, be the number of cases favorable