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## Homework Statement

In a lab experiment, a mouse can choose one of two food types each day, type I and type II. Records show that if a mouse chooses type I on a given day, then there is a 75% chance that it will choose type I the next day and if it chooses type II on one day, then there is a 50% chance that it will choose type II the next day.

(a) If the mouse chooses type I today, what is the probability that it will choose type I two days from now?

(b) If the mouse chooses type II today, what is the probability that it will choose type II three days from now?

## Homework Equations

## The Attempt at a Solution

I think a suitable transition matrix for this phenomenon is:

[tex]Px_{t} = \left[\begin{array}{ccccc} 0.25&0.5 \\ 0.75&0.5 \end{array}\right][/tex] [tex]\left[\begin{array}{ccccc} x_{1}(t) \\ x_{2}(t) \end{array}\right][/tex]

for part (a) I have the initial condition [tex]\left[\begin{array}{ccccc} 1 \\ 0 \end{array}\right][/tex]

[tex]\left[\begin{array}{ccccc} 0.25&0.5 \\ 0.75&0.5 \end{array}\right][/tex] [tex]\left[\begin{array}{ccccc} 2 \\ 0 \end{array}\right][/tex][tex]= \left[\begin{array}{ccccc} 0.5 \\ 1.5 \end{array}\right][/tex]

So the probability is 0.5?

for part (b) the initial condition is (0,1). This time we end up with:

[tex]= \left[\begin{array}{ccccc} 1.5 \\ 2.5 \end{array}\right][/tex] !

The probability of choosing type II in three days is 2.5