Math, linear algebra, electrical engineer, matrix

Question

Hi, I need some help with matrix equations

We will look at a mouse that can move freely between two rooms A and B. If the mouse was most on

room A in the course of one day, it will be most in room B the next day with 80% probability. If

the mouse was most in room B during one of a day, it becomes most in room A the next day with 10%

probability. Let's assume that the mouse is in room A

at the beginning t = 0.

a) Find the transition matrix of this dynamic system and explain what an element in this matrix is

means.

b) What is the probability that the mouse settles in room A or B after t days?

c) If we let the mouse oscillate between rooms A and B until it dies, in which room is it most likely to die and what is the probability of this? Explain how to find the answer from the result in b) without calculating.

Answer 1

Hi Jorgen H.,

My matrix math is rusty so I'll help you with this problem, only. BTW, it's not in the spirit of Wyzant to submit multiple problems on the same topic, same skills, at the same time. Submit one, check periodically if anyone answered, then apply the skills you learned.

But this problem is essentially! qualitative.

Transition matrix (probability of passage, so to speak) is two rows, two columns. Could designate column side as Today, and row side as Tomorrow. Each side has A and B as labels for the two positions. So Row 1 (A label on left) is 0.2 0.8 and Row 2 (B label on left) is 0.1 0.9 . OK so far?

The equilibrium position for the mouse is (1:8) in (A:B) respectively. You may intuitively see this because the quantitative rate of exchange in each direction is therefore equal, which is what equilibrium is (1*0.8 = 8*0.1).

You may model this as an exponential approach to the equilibrium position, starting from A(t=0) of 1.0 . I will leave the interesting details to you, namely, figuring out what the time constant is. I say, "approximation" because the observations are a 'discrete math' process, you did not put the mouse-boxes into a suitcase and close it (as a certain German physicist might have done....food for his cat, perhaps?), to make a continuous function over time; but all you need is the initial and the 1-day point to make your curve! And your "samplings" are exactly at the day intervals.

You can certainly go crazy on part (c). Why would a mouse continue to drag itself between the two compartments, even as death by starvation neared? Couldn't it figure out intuitively that if it preferred compartment B, it might as well hunker there? What does this problem tell us about the potential inhumanity of the question poser? And so on. Have fun!

-- Cheers, --Mr. d.