Computing conditional probability in the Markov chain

[Win_odd Dhamnekar]

Example 4.3 On any given day Hari is either cheerful (C), so-so (S), or glum (G). If he is cheerful today, then he will be C, S, or G tomorrow with respective probabilities 0.5, 0.4, 0.1. If he is feeling so-so today, then he will be C, S, or G tomorrow with probabilities 0.3, 0.4, 0.3. If he is glum today, then he will be C, S, or G tomorrow with probabilities 0.2, 0.3, 0.5. Letting $X_n$ denote Hari’s mood on the nth day, then $\{X_n, n \geqslant 0\}$ is a three-state Markov chain (state 0 = C, state 1 = S, state 2 = G) with transition probability matrix

$\begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S \\ G \end{array} & \begin{Vmatrix} .5 & .4 & 0.1 \\ .3 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.5 \end{Vmatrix}\end{array}$

Question:

In Example 4.3, Hari was in a glum mood four days ago. Given that he hasn’t felt cheerful in a week, what is the probability he is feeling glum today?

My attempt to answer this question:
$$P^4 = \begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S\\ G \end{array} & \begin{Vmatrix} 0.3446 & 0.3734 & 0.282 \\ 0.3378 & 0.3706 & 0.2916 \\ 0.333 & 0.3686 & 02984 \end{Vmatrix} \end{array}$$
That means tomorrow, Hari will be glum with probability 0.2984.

Now, how to compute the probability that today Hari is in glum mood given that he was not cheerful in a week?

Author provided the answer = $$\frac{P^4_{2,2} }{1- P^4_{2,0}}$$ for the markov chain with transition probability matrix $$\begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S\\ G \end{array} & \begin{Vmatrix} 1 & 0 & 0 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.3 & .5 \end{Vmatrix} \end{array}$$
I don't what is the logic behind author's computed answer?

Would any member of free math help forum knows the correct answer with steps to this question?

 

[BigBeachBanana]

Example 4.3 On any given day Hari is either cheerful (C), so-so (S), or glum (G). If he is cheerful today, then he will be C, S, or G tomorrow with respective probabilities 0.5, 0.4, 0.1. If he is feeling so-so today, then he will be C, S, or G tomorrow with probabilities 0.3, 0.4, 0.3. If he is glum today, then he will be C, S, or G tomorrow with probabilities 0.2, 0.3, 0.5. Letting $X_n$ denote Hari’s mood on the nth day, then $\{X_n, n \geqslant 0\}$ is a three-state Markov chain (state 0 = C, state 1 = S, state 2 = G) with transition probability matrix

$\begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S \\ G \end{array} & \begin{Vmatrix} .5 & .4 & 0.1 \\ .3 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.5 \end{Vmatrix}\end{array}$

Question:

In Example 4.3, Hari was in a glum mood four days ago. Given that he hasn’t felt cheerful in a week, what is the probability he is feeling glum today?

My attempt to answer this question:
$$P^4 = \begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S\\ G \end{array} & \begin{Vmatrix} 0.3446 & 0.3734 & 0.282 \\ 0.3378 & 0.3706 & 0.2916 \\ 0.333 & 0.3686 & 02984 \end{Vmatrix} \end{array}$$
That means tomorrow, Hari will be glum with probability 0.2984.

Now, how to compute the probability that today Hari is in glum mood given that he was not cheerful in a week?

Author provided the answer = $$\frac{P^4_{2,2} }{1- P^4_{2,0}}$$ for the markov chain with transition probability matrix $$\begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array}{c c c} C \\ S\\ G \end{array} & \begin{Vmatrix} 1 & 0 & 0 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.3 & .5 \end{Vmatrix} \end{array}$$
I don't what is the logic behind author's computed answer?

Would any member of free math help forum knows the correct answer with steps to this question?

What exactly that you don't understand? The solution is fairly straightforward.

 

[Win_odd Dhamnekar]

What exactly that you don't understand? The solution is fairly straightforward.

Why did author use the modified P(transition probability matrix) instead of original P? According to author's answer $$P^4 = \begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array} {c c c} C \\ S \\ G \end{array} & \begin{Vmatrix} 1 & 0 & 0 \\ 0.7054 & 0.1354 & 0.1593 \\ 0.6522 & 0.1593 & 0.1885 \end{Vmatrix} \end{array}$$ So author's answer is $$\frac{0.1885}{ 1-0.6522} = 0.542 .$$
Firstly,$P^4_{0,1}= 1$ in $P^4$ What does that mean ? It means tomorrow, Hari will be cheerful with probability 1. How did that inference, author seek? In other words, why did author feel the need to modify original transition probability matrix?

Secondly, $P^4_{2,0}$ indicates the probability that tomorrow( not today) , Hari will be glum. Question asked today's probability that Hari will be glum.

Thirdly, how can we know that author's answer is a conditional probability, where the condition is that Hari was not cheerful during the week?

 

[Win_odd Dhamnekar]

Author has not provided any or all of the steps involved in arriving at his answer. Would any member of this forum give all the steps involved in arriving at the author's answer?

 

[Win_odd Dhamnekar]

Sorry, $P^4_{0,0} =1$ in$P^4$

 

[BigBeachBanana]

Why did author use the modified P(transition probability matrix) instead of original P? According to author's answer $$P^4 = \begin{array}{c c} & \begin{array}{c c c} C & S & G \\ \end{array} \\ \begin{array} {c c c} C \\ S \\ G \end{array} & \begin{Vmatrix} 1 & 0 & 0 \\ 0.7054 & 0.1354 & 0.1593 \\ 0.6522 & 0.1593 & 0.1885 \end{Vmatrix} \end{array}$$ So author's answer is $$\frac{0.1885}{ 1-0.6522} = 0.542 .$$
Firstly,$P^4_{0,1}= 1$ in $P^4$ What does that mean ? It means tomorrow, Hari will be cheerful with probability 1. How did that inference, author seek? In other words, why did author feel the need to modify original transition probability matrix?

Secondly, $P^4_{2,0}$ indicates the probability that tomorrow( not today) , Hari will be glum. Question asked today's probability that Hari will be glum.

Thirdly, how can we know that author's answer is a conditional probability, where the condition is that Hari was not cheerful during the week?

Since Hari was not Cheerful during the week, he can't transition from C->S or C->G thus their transition probabilities are 0. That leaves C->C with the probability of 1.

$P^4_{2,0}$ indicates the probability he was in state 2 (Gloomy) 4 days ago and in state 0 today (Cheerful). You're misinterpreting the meaning.

Recall that $P(B|A)=\frac{P(A \cap B)}{P(A)}$.
Define A as feeling Glum 4 days ago and now either feeling So-So or Glum today.
B as feeling Glum 4 days ago and feeling Glum today

P(A)=$P^4_{2,1} + P^4_{2,2}$ = $1-P^4_{2,0}$

The numerator should be intuitive, the probability of feeling Glum 4 days ago and feeling Glum today $P^4_{2,2}$

 

[Win_odd Dhamnekar]

Since Hari was not Cheerful during the week, he can't transition from C->S or C->G thus their transition probabilities are 0. That leaves C->C with the probability of 1.

$P^4_{2,0}$ indicates the probability he was in state 2 (Gloomy) 4 days ago and in state 0 today (Cheerful). You're misinterpreting the meaning.

Recall that $P(B|A)=\frac{P(A \cap B)}{P(A)}$.
Define A as feeling Glum 4 days ago and now either feeling So-So or Glum today.
B as feeling Glum 4 days ago and feeling Glum today

P(A)=$P^4_{2,1} + P^4_{2,2}$ = $1-P^4_{2,0}$

The numerator should be intuitive, the probability of feeling Glum 4 days ago and feeling Glum today $P^4_{2,2}$

I understood all the steps author took to arrive at his answer after careful thinking.. Author's answer is correct. You answered all my queries satisfactorily.