AppendixI.tex

\section{Appendix I: Proving Markov Models Work on 2 by 2 Matricies}
\subsubsection{Assumptions}

Let $A \in \mathbb{R}^{2\times2}$

$A =
\begin{pmatrix}
  a & b\\
  c & d\\
\end{pmatrix}
$

$a+b=1$\\
$c+d=1$\\
$a$ is the probability of moving from state $O$ to state $O$ in one turn.\\
$b$ is the probability of moving from state $O$ to state $T$ in one turn.\\
$c$ is the probability of moving from state $T$ to state $O$ in one turn.\\
$b$ is the probability of moving from state $T$ to state $T$ in one turn.

Let $XY_n$ be the event of moving from state $X$ to $Y$ in $n$ turns. \\
Let $P(X_n)$ be the probability of being in state $X$ on turn $n$.\\
Let $X$,$Y$ and $Z$ be generically either states $O$ or $T$.

\subsection{Derivation}

(1) $P(XY_2) = P(OY_1) \times P(O_1|X_0)+P(TY_1) \times P(T_1|X_0)$\\
(2) $P(Z_1|X_0) = P(XZ_1)$\\
(3) $P(XY_2) = P(OY_1) \times P(XO_1)+P(TY_1) \times P(XT_1)$

This general formula can be used to find the odds of going from state to state in 2 turns.\\
$P(OO_2) = P(OO_1) \times P(OO_1)+P(OT_1) \times P(TO_1) = a \times a + b \times c$\\
$P(OT_2) = P(OO_1) \times P(OT_1)+P(OT_1) \times P(TT_1) = a \times b + b \times d$\\
$P(TO_2) = P(TO_1) \times P(OO_1)+P(TT_1) \times P(TO_1) = c \times a + d \times c$\\
$P(TT_2) = P(TO_1) \times P(OT_1)+P(TT_1) \times P(TT_1) = c \times b + d \times d$
\[
A^2 =
\left(
\begin{array}{c c}
a \times a+b \times c & a \times b+b \times d \\
a \times c+c \times d & b \times c+d \times d \\
\end{array}
\right)
=
\left(
\begin{array}{c c}
P(OO_2) & P(OT_2) \\
P(TO_2) & P(TT_2) \\
\end{array}
\right) \]