SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

40 I. R. Shafarevichbility will be between 148 000 and 152 000 (since 3 4 n = 150 000, n 1100 = 2000,np ; n" = 148 000, np + n" = 152 000).Conversely, using Chebyshev's theorem we can estimate the number of experimentsto be made in order to obtain the probability p accurately enough. Supposethat we want to determine it with accuracy up to 1=10 and that the probability itis equal to the obtained number is not less than 0,99. According to Chebyshev'stheorem we have toput" =1=10 and to use the inequality; 110pq 2 n < 001:Notice that q =1;p, and for any p such that 0 6 p 6 1, wehave pq = p(1;p) 6 1=4.This follows from the fact that the geometric mean is not greater than the arithmeticmean of the numbers p, q, whichis1=2. Therefore, it is enough that n should satisfythe inequality14; 110 2 n < 001which implies n>2500.Problems1. In the set of some objects, 95% of them have a certain property. Prove thatamong 200 000 objects, the number of those which have thisproperty isbetween189 000 and 191 000 with probability not less than 0,99.2. Modify Problem 1 so that the portion of objects which have a certainproperty isnotknown. What is the probability that after testing 100 objects wecan determine it with accuracy up to 0,1?3. For any positive integer r 6 n nd the sum of all termsk(k ; 1)(k ; r +1)p(A k )for k =1 ...n.4. For r 6 4 evaluate the sums r consisting of terms k r p(A k ) for all k =0 1 2 3 4. Do this in two dierent ways: a) by the reasoning of the proof of theLemma, and b) by expressing the sums r in terms of sums evaluated in Problem 3for r =1 2 3 4.5. Try to improve the inequality (4) in Chebyshev's theorem, applying the k ; np 4 kfactorinstead of; np 2.The improvement will be that n2 willn"n"appear in the denominator of the right-hand side of the inequality instead of n.I. R. Shafarevich,Russian Academy of Sciences,Moscow, Russia