Intro to Markov Chains

1. Sep 28, 2010

jumbogala

1. The problem statement, all variables and given/known data
On any given day, Amy is either cheerful (C), neutral (N), or mad (M). If she is cheerful today, then she will be C, N, or M tomorrow with probabilities 0.5, 0.4, 0.1 respectively. The rest of the probabilities are given so that we can get the probability matrix P:

P = [
0.5 0.4 0.1
0.3 0.4 0.3
0.2 0.3 0.5
]

Amy is currently in a cheerful mood. What is the probability that she is not in a mad mood on any of the following three days?

2. Relevant equations

3. The attempt at a solution
I'm not really sure how to go about this. To find the probability that Amy would be mad in 3 days, I would find P3 and look at the (1,3) entry, correct?

Then to find the probability that she is NOT mad in 3 days, I'd take 1 minus that probability.

But it doesn't ask for the 3rd day, it asks for "any of the following 3 days". I'm confused about how to do that!

The solutions manual changes the matrix to
[
0.5 0.4 0.1
0.3 0.4 0.3
0.0 0.0 1.0
]

Raises that to the power of 3, then takes 1 - [the (1,3) entry of the matrix]

I don't get why. Thanks!