Assume we have two slot machines labeled M1 and M2, each producing a sequence of 3 binary digits. After running the machines, we display the decimal representation of the binary sequence produced by each machine. For example, if we observe 111, we note the number 7. The machines operate the following way:
— M1 produces, with equal probability, all the possible sequences of 3 binary digits. So the possible outputs of the machine are in the set $\{0,1,...,7\}$.
— M2 behaves as M1, except that it has its first digit blocked to 0. So it can output only from the set $\{0, 1, 2, 3\}$.
You pick one of the machines at random with equal probability and run it indefinitely. The sequence obtained can be modeled as the output of the source $S = S_1,S_2,...$ where $S_i \in \{0,1,...,7\}$.
How can I compute $$\mathbb P_{S_1,...,S_n}(s_1,...,s_n)\ \ ?$$
Hint: $\mathbb P_{S_1,...,S_n}(s_1,...,s_n) = \sum_{i=1}^{2} 0.5*\mathbb P_{S_1,...,S_n}(s_1,...,s_n | M_i) = \sum_{i=1}^{2} 0.5* \prod_{j=1}^{n} \mathbb P_{S_j}(s_j| M_i),$
where the first equality follows from Law of Total probability and the last equality is "assumed" to be true from the information given since sequences $S_j$'s do not depend on each other.
Further $\mathbb P_{S_j}(s_j| M_1) = \frac{1}{8}$ and $\mathbb P_{S_j}(s_j| M_2) = \frac{1}{4}.$