Probability
# Markov Chains

$($start, end$)$ is most likely?

Suppose a process begins at one state and progresses to another state after 3 turns. Which pairA Markov chain has transition matrix $\begin{pmatrix} 0.5 & 0.3 & 0.2 \\ 0.4 & 0.4 & 0.2 \\ 0.6 & 0.4 & 0 \end{pmatrix}.$

What is its $2$-step transition matrix?

**(A)**: $\begin{pmatrix}
0.49 & 0.35 & 0.16 \\
0.48 & 0.36 & 0.16 \\
0.46 & 0.34 & 0.2
\end{pmatrix}$

**(B)**: $\begin{pmatrix}
0.25 & 0.35 & 0.4 \\
0.39 & 0.25 & 0.36 \\
0.36 & 0.39 & 0.25
\end{pmatrix}$

**(C)**: $\begin{pmatrix}
0.49 & 0.35 & 0.16 \\
0.48 & 0.36 & 0.16 \\
0.48 & 0.32 & 0.2
\end{pmatrix}$

**(D)**: $\begin{pmatrix}
0.49 & 0.35 & 0.16 \\
0.4 & 0.44 & 0.16 \\
0.48 & 0.32 & 0.2
\end{pmatrix}$

Phineas is flipping a fair coin repeatedly and marking down the results. However, he has a soft spot for heads, so half the time he marks heads, he marks heads again. (This means he could possibly mark down heads many times in a row without actually flipping the coin again.) The resulting sequence of states could be modeled by a Markov chain, where the first possible state (and row of the matrix) is heads and the second possible state is tails. What is its transition matrix?

**(A)**: $\begin{pmatrix}
\tfrac{1}{2} & \tfrac{1}{2} \\
\tfrac{1}{2} & \tfrac{1}{2}
\end{pmatrix}$

**(B)**: $\begin{pmatrix}
\tfrac{1}{2} & \tfrac{1}{2} \\
\tfrac{1}{4} & \tfrac{3}{4}
\end{pmatrix}$

**(C)**: $\begin{pmatrix}
\tfrac{3}{4} & \tfrac{1}{2} \\
\tfrac{1}{4} & \tfrac{1}{2}
\end{pmatrix}$

**(D)**: $\begin{pmatrix}
\tfrac{3}{4} & \tfrac{1}{4} \\
\tfrac{1}{2} & \tfrac{1}{2}
\end{pmatrix}$

A Markov chain can be constructed for the weather based on the following table.

States | Rainy Tomorrow | Cloudy Tomorrow | Sunny Tomorrow |

Rainy Today | $\tfrac{1}{2}$ | $\tfrac{1}{3}$ | $\tfrac{1}{6}$ |

Cloudy Today | $\tfrac{1}{3}$ | $\tfrac{1}{3}$ | $\tfrac{1}{3}$ |

Sunny Today | $0$ | $\tfrac{1}{9}$ | $\tfrac{8}{9}$ |

If it is rainy today, what is the probability that it will be rainy in three days?