[![HMM model][1]][1]


This is the HMM model considered in the question
[![Emission probability matrix][2]][2]


And this is the emission probabilities for the respective states. There are two emission values, bringing an umbrella and not bringing an umbrella.

**Description of the question:**

The question considered the *initial state as sunny*.
On *day 2*, the person *brought an umbrella*. 
On *day 3*, the person *didn't carry an umbrella*.

We have to find the probability that it's foggy on 3rd day.

Given: Prior probability of the caretaker carrying an umbrella is 0.5.

According to me, the answer to this question was addition of the terms in the red circles.

Where &alpha;<sub>1</sub> = 0.8 * 0.1, &alpha;<sub>2</sub> = 0.05 * 0.8, &alpha;<sub>3</sub> = 0.15 * 0.3

**Problem I faced:**
But the answer included dividing the answer I got by (0.5)<sup>2</sup>. That is, I guess they are dividing by the prior probability two times for the two transitions, but I'm not clear as to why they are dividing.
[![Answer][3]][3]


  [1]: https://i.sstatic.net/E3u45.jpg
  [2]: https://i.sstatic.net/XDGFn.jpg
  [3]: https://i.sstatic.net/QklOg.png