Abstract Algebra Theory and Applications - Computer Science ...

7.1 ERROR-DETECTING AND CORRECTING CODES 113Theorem 7.1 If a binary n-tuple (x 1 , . . . , x n ) is transmitted across a binarysymmetric channel with probability p that no error will occur in each coordinate,then the probability that there are errors in exactly k coordinates is( ) nq k p n−k .kProof. Fix k different coordinates. We first compute the probability thatan error has occurred in this fixed set of coordinates. The probability of anerror occurring in a particular one of these k coordinates is q; the probabilitythat an error will not occur in any of the remaining n − k coordinates is p.The probability of each of these n independent events is q k p n−k . The numberof possible error patterns with exactly k errors occurring is equal to( nk)=n!k!(n − k)! ,the number of combinations of n things taken k at a time. Each of theseerror patterns has probability q k p n−k of occurring; hence, the probability ofall of these error patterns is( ) nq k p n−k .kExample 4. Suppose that p = 0.995 and a 500-bit message is sent. Theprobability that the message was sent error-free isp n = (0.995) 500 ≈ 0.082.The probability of exactly one error occurring is( ) nqp n−1 = 500(0.005)(0.995) 499 ≈ 0.204.1The probability of exactly two errors is( ) nq 2 p n−2 500 · 499= (0.005) 2 (0.995) 498 ≈ 0.257.22The probability of more than two errors is approximately1 − 0.082 − 0.204 − 0.257 = 0.457.□