A source emits symbols 0 and 1 for transmission to a receiver. Random noises \(S_{1}, S_{2}, \ldots\) successively and independently affect the transmission process of a symbol in the following way: if a ' 0 ' ('1') is to be transmitted, then \(S_{i}\) distorts it to a '1' (' 0 ') with probability \(p(q) ; i=1,2, \ldots\) Let \(X_{0}=0\) or \(X_{0}=1\) denote whether the source has emitted a ' 0 ' or a ' 1 ' for transmission. Further, let \(X_{i}=0\) or \(X_{i}=1\) denote whether the attack of noise \(S_{i}\) implies the transmission of a ' 0 ' or a ' 1 '; \(i=1,2, \ldots\). The random sequence \(\left\{X_{0}, X_{1}, \ldots\right\}\) is an irreducible Markov chain with state space \(\mathbf{Z}=\{0,1\}\) and transition matrix

\[\mathbf{P}=\left(\begin{array}{cc} 1-p & p \\ q & 1-q \end{array}\right)\]

(1) Verify: On condition \(0

\[\mathbf{P}^{(m)}=\frac{1}{p+q}\left(\begin{array}{ll} q & p \\ q & p \end{array}\right)+\frac{(1-p-q)^{m}}{p+q}\left(\begin{array}{cc} p & -p \\ -q & q \end{array}\right)\]

(2) Let \(p=q=0.1\). The transmission of the symbols 0 and 1 is affected by the random noises \(S_{1}, S_{2}, \ldots, S_{5}\). Determine the probability that a ' 0 ' emitted by the source is actually received.