Prior probability in HMM
Let $s_2,s_3$ be the states on the second and third days, and let $x_2,x_3$ be the indicators of carrying an umbrella on these days. Then
$$
\newcommand{\Sunny}{\mathit{Sunny}}
\newcommand{\Rainy}{\mathit{Rainy}}
\newcommand{\Foggy}{\mathit{Foggy}}
\newcommand{\Yes}{\mathit{Yes}}
\newcommand{\No}{\mathit{No}}
\Pr[s_3 = \Foggy \mid s_1 = \Sunny, x_2 = \Yes, x_3 = \No] =
\frac{\Pr[s_3 = \Foggy, x_2 = \Yes, x_3 = \No \mid s_1 = \Sunny]}{\Pr[x_2 = \Yes, x_3 = \No \mid s_1 = \Sunny]}
$$
It seems you might have forgotten the denominator.