Probability of Type I/Type II Selection in Lab Experiment

[roam]

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:

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

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

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

So the probability is 0.5?

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

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

The probability of choosing type II in three days is 2.5

 

[roam]

And btw last part of the questions asks:

If there is 10% chance that the mouse will choose type I today, what is the probability that it will choose type I tomorrow?

I'm not sure how to use my matrix to solve find this.
I appreciate some guidance. Thanks :)

 

[Borek]

Isn't your matrix transposed?

 

[roam]

Isn't your matrix transposed?

No, which matrix?

$$\left[\begin{array}{ccccc} x_{1}(t) \\ x_{2}(t) \end{array}\right]$$ is the state vector.

 

[Mark44]

I think Borek meant your transition matrix.

 

[Borek]

I think that's what I thought. Rows should sum to 1.

 

[roam]

I'm looking at an example in my textbook and only columns sum to 1 not rows.

 

[Borek]

So perhaps you should use a row vector for a state vector? That's a matter of convention.

Sum of probablities should be 1, so both your state vectors (for a and b) are wrong.