Probability of Amy Not Being Mad in the Next Three Days

[jumbogala]

Homework Statement

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?

Homework Equations

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!

 

[mighty2000]

I would approach this problem by using the principles of probability and conditional probability. First, I would rewrite the given information in a more clear and organized form:

Current Mood: Cheerful (C)

Transition Probabilities:
C -> C: 0.5
C -> N: 0.4
C -> M: 0.1

N -> C: 0.3
N -> N: 0.4
N -> M: 0.3

M -> C: 0.2
M -> N: 0.3
M -> M: 0.5

Next, I would use the principle of conditional probability to calculate the probability of Amy being in a particular mood on any given day. For example, the probability of Amy being in a cheerful mood on the second day (assuming she was cheerful on the first day) would be:

P(C on day 2) = P(C -> C) * P(C on day 1) + P(C -> N) * P(N on day 1) + P(C -> M) * P(M on day 1)
= (0.5 * 1) + (0.4 * 0) + (0.1 * 0)
= 0.5

Similarly, the probability of Amy being in a neutral mood on the third day (assuming she was cheerful on the first day) would be:

P(N on day 3) = P(N -> C) * P(C on day 1) + P(N -> N) * P(N on day 1) + P(N -> M) * P(M on day 1)
= (0.3 * 1) + (0.4 * 0) + (0.3 * 0)
= 0.3

Now, to answer the question of the probability of Amy not being in a mad mood on any of the following three days, we can use the principle of complement probability. This means that the probability of an event not happening is equal to 1 minus the probability of the event happening. In this case, the event is Amy being in a mad mood on any of the following three days. Therefore, the probability we are looking for is:

P(not mad in 3 days) = 1 - P(mad in 3 days)

To calculate the probability of Amy being mad in 3 days, we can use