Moreover, the event is certain to happen. Certainty is therefore expressed by 1. If there are no favorable cases, the event is impossible, so that the expression for impossibility is '-^- Two events are said to be complementary if the happening of one excludes the possibility of the other, and the sum of their probabilities is 1. For example, if we draw a ball from a bag containing four white and two black balls, the probability of drawing a white ball is ^ and the probability of drawing a black ball is -. Moreover, the two events cannot happen at the same time. They are therefore complementary. THEOREM. Tf the probability that an event will happen isp, the probability that it will fail is 1—p. Proof. Let q be the probability of failure. Then a -, b p= ——, and q = —— 1 /Tf _L A /Tf-L/1
P — ————T dii^i v — , - a+b ' a+6 The happening and the failure of the event are then comple- mentary, since they are mutually exclusive, and ia b , ^ P+q=——,+—,-i=1; - a+b a+b consequently, q=l—p, (2) as was to be proved. 65. Simple problems in probability. Many of the simpler prob- lems in the theory of probability may be solved by means of the definition of probability, and the fundamental theorems and for- mulas from the theory of permutations and combinations. The truth of the following propositions concerning permutations and combinations will be assumed. 178 MATHEMATICAL THEORY OF INVESTMENT 1. If one act can be performed in p ways, and if, after this act is completed, a second unrelated act can be performed in q ways, the number of ways in which the two acts can be performed in succession is pq. 2. The number of permutations of n things taken r at a time is ^-EF w andfor the special case where n=r 4.=[n. (2) 3. The number of combinations of n things taken r at a time is ^o=—1— (3) " \r\n—r EXAMPLES 1. A die is thrown once. What is the probability that the number of points is less than 5 ? Suggestion. There are four favorable cases. 2. Ten balls, exactly alike except that they are numbered from 1 to 10, are put into a bag, and a single ball is drawn at random. What is the proba- bility that the ball is numbered 1 ? What is the probability that the ball is numbered either 1 or 2 ? 3. If two of the balls described in Example 2 are drawn simultaneously, what is the probability of drawing the pair numbered 3 and 5 ? Suggestion. There are C'(1,0) = 45 ways of selecting a pair from 10 numbered balls. 4. A bag contains n balls numbered consecutively from 1 to n, and from it three are drawn simultaneously at random. What is the probability that the numbers are 1,2, and 3 ? 5. Two coins are tossed into the air at the same time (or in succession). What is the probability that both will fall heads? Suggestion. By proposition 1, two coins may fall in any one of four ways. 6. What is the probability that 10 coins tossed into the air at the same time will all fall heads? What is the probability that a single coin tossed into the air 10 times will fall heads every time ?