What's the probability that the teacher teaches her class?

You say that the probability that a given student shows up and the weather is bad is $\Pr(B)p_b$. This is correct. However, you go on to say, that the probability that $j$ given students show up and the weather is bad is $(\Pr(B)p_b)^j$. This is incorrect when $j>1.$ The correct value is $\Pr(B)p_b^j$. After all, the weather is only bad on one day, not on $j$ days. We have $j+1$ events: the weather is bad, and $j$ students show up.

I would break it up a little differently. Let's use the law of total probability:

$P\left(\text{teaches}\right) = P\left(\text{teaches}~|~\text{good weather}\right)P\left(\text{good weather}\right) + P\left(\text{teaches}~|~\text{bad weather}\right)P\left(\text{bad weather}\right)$

$P\left(\text{teaches}\right) = P\left(\text{teaches}~|~\text{good weather}\right)\left(1 - P\left(B\right)\right) + P\left(\text{teaches}~|~\text{bad weather}\right)P\left(B\right)$

Now we only need to calculate two things:

$P\left(\text{teaches}~|~\text{good weather}\right)$ is, conceptually the "probability k or more students show up in good weather", which is given by the binomial formula:

$\displaystyle\sum_{i=k}^{n}{n \choose i}p_g^i(1-p_g)^{n-i} $

And the same for "bad weather", but with $p_b$.