#posterior-state

Question 1: Single choice

A two-state HMM starts in state 1 or 2 with probability $1/2$ each. State 1 stays in state 1 with probability 0.99 and moves to state 2 with probability 0.01; state 2 never leaves. Symbol 3 is emitted with probability 0.8 in state 1 and 0.7 in state 2. The HMM runs for millions of steps, and you see only the last output symbol, which is 3. What is the most likely hidden state at that last step?

State 1, because it emits 3 more often than state 21
Both are equally likely, because the HMM started in each with probability $1/2$ 2
It cannot be decided from the last symbol alone3
State 2, because state 1 has almost surely been left long before4

Timed test