THE THEORY OF PROBABILITY 183 68. Probability of an event when several trials are made. THEOREM I. The probability that an event will happen exactly r times in n trials is w(n-l)... (TO-r+1) 1.2.3...r pf ' where p is the probability that it will happen and q the probability that it will fail in a single trial. Proof. By (4) of § 67 the compound probability that the event will happen in a given trial and fail in the other n — 1 is pf-1. The total probability that it will happen in some one of the n trials is the sum of the probabilities that it will happen in the separate trials, viz. p(f~l+pq''-l+ to n terms = np<f~1. Again, the compound probability that an event will happen in two assigned trials and fail in the other n — 2, say the fifth and the eleventh, is »ft^-t r 1 i and the total probability that it will happen in any two trials whatever is ^"-2+^-2^. . . ^ where the number of terms is equal to the number of ways of specifying two trials out of n, i.e. CQ). But, by proposition <3)of§65' ^ .(.-1) ^—Y^-' m (' V) —— I ~\ Therefore the number of terms is —',——L, and consequently the total probability that an event will happen twice and fail n — 2 times in n trials is ^-1).^-.
1.2
pY-
In general, the probability that the event will happen in r assigned trials and fail in the other n — r trials is ^Y-. 184 MATHEMATICAL THEORY OF INVESTMENT But there are _ n(n-l)(M-2)---(n-r+l) cu- 1.2.3...r ways of specifying r trials out of n trials. The total probability that the event will happen exactly r times in n trials is p'qn-r +p'qn-' + ., to C'(^) terms. It is therefore cwf-^-^-^-^r-f-. THEOREM II.The probability that an event will happen at least r times in n trials is ^-^-^p-'f^ - ^^::;^i"f- where p is the probability that the event will happen, and q the probability that it will fail in a single trial. The event will happen r times if it happens n times, or if it happens n — 1 times, or if it happens n — 2 times, and so on to n — (n — r) times. Consequently, the required probability is the total probability made up of the partial probabilities that it will happen n times, n — 1 times, and so on. But, by Theorem I, the partial probability that the event will happen n times in n trials is j>"; that it will happen n—1 times is np"~lq, that it will happen n—2 times is -—o—^""2?2; and, finally, the proba- n (n — V) (r +1') bility that it will happen r times is ———„'——-^——- p'q"'r. J_ A * o \^n — r) Consequently, the total probability that it will happen at least r times is w,+"<^,--v+ ^y^:::^^-'- EXAMPLES 1. Find the probability of throwing 1 and only 1 point in two trials with 1 die. 2. Find the probability of throwing at least 1 point in 2 throws of 1 die. 3. If, on an average, 99 out of 100 ships reach port safely, find the probability that at least 2 out of 10 will arrive.