LS book (Recursive Macro Theory). Chapter8: Complete Market
Since this is a Markov chain

\begin{eqnarray}
\pi_t(0,0,\cdots,1,1) &amp;=&amp;  \color{blue}{\pi(s_t = 0 | s_{t - 1} = 0)} \cdots \color{magenta}{\pi(s_2 = 0 | s_{1} = 1)} \color{red}{\pi(s_1 = 1 | s_{0} = 1)}\color{orange}{\pi(s_0 = 1)} \\
&amp;=&amp; \color{blue}{1} \times \cdots\times \color{magenta}{0.5}\times\color{red}{1}\times \color{orange}{1} = 0.5
\end{eqnarray}

But you are right if $t &gt; 2$ then $\pi(s_{t + 1}  = 0 | s_{t} = 0) = 1$, so if $s_t = 0$ then $s_{t + 1} = 0$, but at $t = 2$ there's a 50/50 chance that the state changes from $1$ to $0$, so the state may remain $1$ as in the first history, or switch to $0$ as the second one