Hidden Markov Model

Consider the following HMM, which models an intronic sequence with GC rich regions.

The model consists of $9$ states:

The states and transition probabilities are indicated in the following diagram:

Fig. $1.$ The states and transition probabilities.

The emission probabilities for the non$-$silent states are as follows:

Base S$1$ S$2$ S$3$ S$4$ S$5$ S$6$ S$7$

A $00\mathrm{.}50\mathrm{.}250\mathrm{.}400\mathrm{.}2800\mathrm{.}40$

C $000\mathrm{.}250\mathrm{.}150\mathrm{.}220\mathrm{.}200\mathrm{.}10$

G $100\mathrm{.}200\mathrm{.}300\mathrm{.}050\mathrm{.}500\mathrm{.}50$

T $00\mathrm{.}50\mathrm{.}300\mathrm{.}150\mathrm{.}450\mathrm{.}300$

Please answer the following questions:

i$.$ What is the probability of observing GATTAG along the state path p $=$STARTS$1-$S$3-$S$2-$S$4-$S$5-$S$7-$END?

ii$.$ What is the probability of seeing GTAGA, given the current HMM? $($Show the

steps of your calculation.$)$

iii. What is the probability of seeing GTACCG, given the current HMM? State

your observation