In November 2012 Washington joined Colorado in legalizing recreational marijuana. Voters passed ballot Initiative 502 with 56 percent in support amid an 81 percent voter turnout at the polls. The proposal removed most state prohibitions on marijuana manufacture and commerce, permitted limited marijuana use for adults aged 21 and over, and established the need for a licensing and regulatory framework to govern the state’s marijuana industry. Initiative 502 further imposed a 25 percent excise tax levied three times (on marijuana producers, processors, and retailers) and earmarked the revenue for research, education, healthcare, and substance-abuse prevention, among other purposes. 24

Where residual variances are not explicitly included, or as a more general solution, at any change of direction encountered in a route (except for at two-way arrows), include the variance of the variable at the point of change. That is, in tracing a path from a dependent variable to an independent variable, include the variance of the independent-variable except where so doing would violate rule 1 above (passing through adjacent arrowheads: ., when the independent variable also connects to a double-headed arrow connecting it to another independent variable). In deriving variances (which is necessary in the case where they are not modeled explicitly), the path from a dependent variable into an independent variable and back is counted once only.

At the end of 2 nd week the state vector is
Px 1

x (2) =
Px (1)
=
|.25
.20
.25
.30|
|.25|
=
|.2550 |
|.20
.30
.25
.30|
|.20|
=
|.2625 |
|.25
.20
.40
.10|
|.25|
=
|.2325 |
|.30
.30
.10
.30|
|.30|
=
|.2500 |

Note that we can compute x 2 directly using
x 0 as
x (2) = Px (1) = P(Px (0) ) =
P 2 x (0)
Similarly, we can find the state vector for 5 th , 10 th , 20 th , 30 th , and 50 th observation periods.
x (5) =
P 5 x (0) =
.2495
.2634
.2339
.2532
x (10) =
P 10 x (0) =
.2495
.2634
.2339
.2532
x (20) =
P 20 x (0) =
.2495
.2634
.2339
.2532
x (30) =
.2495
.2634
.2339
.2532

x (50) =
.2495
.2634
.2339
.2532

The same limiting results can be obtained by solving the linear system of equations P P = P using this JavaScript. It suggests that the state vector approached some fixed vector, as the number of observation periods increase. This is not the case for every Markov Chain. For example, if
P =
0
1
1
0
, and