China Mathematical Competition

Question

An information station employs four different codes, A, B, C and D, for communication, but each week uses only one of them. The code used in a definite week is randomly selected with equal chance among the three ones that have not been used in the last week. Suppose the code used in the first week is A. Then the probability that A is also used in the seventh week is _______. (expressed as an irreducible fraction)

Accepted Answer

Let P_k denote the probability that code A is used in the kth week. Then the probability that A is not used in the kth week is 1 - P_k. Therefore, we have P_k+1 = 13(1 - P_k). Or P_k+1 - 14 = -13(P_k - 14). As P_1 = 1, P_k - 14 is then a geometric sequence with 34 as the first term and -13 as the common ratio. So we have P_k - 14 = 34(-13)^k-1. Or P_k = 34(-13)^k-1 + 14. Therefore, P_7 = 61243.