User:Aiden Fisher/Sandbox

List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
Binary	binary
Decimal	decimal
Hexadecimal	hexadecimal
Continued fraction	cfrac
Algebraic form	aform

Expected finishing time

The initial state of the system at time zero is state 2, which we represent by the probability vector v:

$\mathbf {v} ={\begin{bmatrix}0&1&0&0&0\end{bmatrix}}.$

The probability distribution of the states at time k is then given by

$\mathbf {p} _{k}=\mathbf {v} P^{k}\,,$

or alternativly,

$\mathbf {p} _{k}=\mathbf {p} _{k-1}P\,.$

The game ends when we reach state 5. So if we define $\mathbf {e} _{5}$ as a column vector of zeros with 1 in the fifth entry the probability that the game has ended at time $k$ is,

$p_{k}=\mathbf {v} P^{k}\mathbf {e} _{5}\,.$

The expected time until the game has ended is K,

$E[K]=\sum _{k=0}^{\infty }k(1-p_{k})=\sum _{k=0}^{\infty }k(1-\mathbf {v} P^{k}\mathbf {e} _{5})=4.5$