The elements of probability (The Addison-Wesley series in by Simeon M Berman

By Simeon M Berman

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N ~ 1) qn-lpO + (n ~ 1) qn-2pl + ... n-- l1) q p + ... + (nn-- l1) qp° + (k n-k k-l n-l · Reason. Apply the Binomial Formula with q, p, n - I, and k - 1 in place of A, B, N, and s, respectively. 28 THE LA\\! -I = k (n) k PJ,. ,. Reasoll. ith 11, k in place of N, s, respectrvely. 4. 1). Reason. 1). 5. The assertion of the proposition follows. Reason. Combine Statements 1 and 4. 2. 5. 2 A drug which is 35 percent effective is taken by 20 patients. 35) = 7. 3 The expected number of correctly called cards in the ESP experiment is 2S(t) or 5, under the hypothesis of no ESP (random guessing).

Here are some of the rules used for such computations: + + + + 1. The area under the curve to the right (or to the left) of the point x = 0 is ! because the curve is symmetric about 0 and the total area is 1. 2. The area to the right of a positive number z is the difference between the area to the right of x = 0 (that is, i) and the area from x = 0 to x = z, which is given in the table. 3. 2 above. 4. The area to the left of a positive number x = z is equal to area from 0 to z. i plus the 47 THE NORMAL APPROXIMATION 5.

The proof is outlined in Exercise 7 below; applications are contained in several other exercises. 3,2 EXERCISES 1. Find the limit L of each sequence, stating which definition has been employed: + cw + + a) Xl = (t) 3, X2 = 3, . , . • b) Xl = (t) - 3, X2 = (~yl - 3, ... , Xn = (t)n - 3, ... 1)2, ... 1)n, ... d) Xl = 1, X2 = -i, . , Xn = (-i)n, .... e) Xl = 2, X2 = 2, ... , Xn = 2, .. , . 2. 333 ... 4. 5, respectively, is also true. 4. In algebra it is shown that the value of the infinite geometric series 1 is equal to 1/(1 - r) if -1 < I' that the sequence of partial sums 1,1 + r + r2 + ...

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