# LS book (Recursive Macro Theory). Chapter8: Complete Market

I am reading LS book, on Chapter 8 - Complete Markets (3rd version). There is an example on page 264 that I am quite confused.

I attach the picture of the example here under.

I don't understand why we have a history (0,0,0,...,0,1,1)? since if $$s_t=0$$, then $$s_{t+1} = 0$$ for sure, right?

Or the authors means the history here is in order $$(s_t, s_{t-1},....,s_1,s_0)$$, so it makes sense since we know that $$s_0=1, s_1=1$$, but isn't it quite weird, since we usually write a history $$h_t=(s_0,s_1,...s_t)$$ right?

Anyone has an idea? Really appreciate your help!

$$\begin{eqnarray} \pi_t(0,0,\cdots,1,1) &=& \color{blue}{\pi(s_t = 0 | s_{t - 1} = 0)} \cdots \color{magenta}{\pi(s_2 = 0 | s_{1} = 1)} \color{red}{\pi(s_1 = 1 | s_{0} = 1)}\color{orange}{\pi(s_0 = 1)} \\ &=& \color{blue}{1} \times \cdots\times \color{magenta}{0.5}\times\color{red}{1}\times \color{orange}{1} = 0.5 \end{eqnarray}$$
But you are right if $$t > 2$$ 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