From weather conditions to baseball scores to stock performances, many probabilistic real-world systems can be modeled with Markov chains.

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

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

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?

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