# Markov Chains

## 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?

## 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

## Answers and Replies

Related Precalculus Mathematics Homework Help News on Phys.org
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
Mentor
Isn't your matrix transposed?

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
Mentor
I think Borek meant your transition matrix.

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

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

Borek
Mentor
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.