I wake up in a random class and hear 6 biology-related words. How certain should I be that I'm in Biology class?

Yes, your reasoning is correct... the posterior probability for each update becomes the prior probability for the next. (This is one of the nice things about the Bayesian approach.) Note that each update can be written as $$ P' = \frac{p_w P}{p_w P + q_w (1-P)}=\frac{p_w P}{q_w + (p_w - q_w)P}=\frac{P}{\alpha_w +(1-\alpha_w)P}, $$ where $\alpha_w=P(w|\neg bio) \div P(w|bio)$ is $5/6$ or $6/7$ or $7/8$ for your words. It's easy to check that the result after all six words comes out to $P\approx 0.221453$, and that this is independent of the order in which you do the updates.

In light of the other answer, it's worth noting that this is the same as the result from a single update with $\alpha=\prod_w \alpha_w=25/64$... that is, it's the same as treating the words as independent. This is exactly what the diagram says: the six words are independent, given the class. The advantage of the first approach, though, is that you can update your credence in an online fashion as you hear the words... allowing you to, say, take out your textbook as soon as you're sufficiently confident you're in the right class.