Delete this post! I double posted!
sorry about your diarrhea
$\begin{align*}
&P[\text{winning tommorow}] = \\
&P[\text{winning tomorrow | tomorrow is calm}]P[\text{tomorrow is calm}]+ P[\text{winning tomorrow | tomorrow is bad}]P[\text{tomorrow is bad}] = \\
&(0.7)(1-0.3) + (0.5)(0.3) = \\
&0.49 + 0.15 = 0.64
\end{align*}$
Tomorrow has a 70% chance of being F(fair weather. WHY?
If $B~\&~W$ stand for "bad weather" and "winning" resp.
$\mathcal{P}(W)=\mathcal{P}(W\cap F)+\mathcal{P}(W\cap B)=\mathcal{P}(W|F)\mathcal{P}(F)+\mathcal{P}(W|B) \mathcal{P}(B)$
$\mathcal{P}(W|B)\mathcal{P}(B)=(0.5)(0.3)$ WHY?
$\mathcal{P}(W|B)$ stands for "probability of winning given bad weather".