Simple Bayesian classification has me discouraged 
My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has. I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info.
My understanding from stats way back when is that p(x|ω1)
p
(
x
|
ω
1
)
 is stating the probability of x
x
 given ω1
ω
1
for 0≤x≤2
0
≤
x
≤
2
 and zero other wise. Same obviously if given ω2
ω
2
. If I understand, the priors are the probability of ω1
ω
1
 and ω2
ω
2
 which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated.
 A: Bayesian Decision Theory is your way to go for googling. For a quick setup, I can say that we are interested in devising strategies for classifying an incoming sample, namely $x$ optimally. Here, Bayesian Decision Theory looks at posterior distributions, i.e. choose the class with the highest posterior: $p(\omega_i|x)$, where $i \ \ \epsilon \ \ \{1,2\}$. The equality holds at a specific $x$ value, and that is what option a is asking.
A: I am a little surprised that the relevant material for this question wasn't covered in the lectures prior to asking this question. If that is the case, it is pretty unreasonable on the part of your university to expect students to answer this question. 
I would suggest reading Chapter 2 on "Bayesian Decision Theory" in Pattern Classification by Duda, Hart and Stork. On the other hand, these set of slides [A] feel reasonably complete. You can, of course, find more material on Bayesian Decision Theory by doing a quick internet search. For the concept of Bayesian Decision Boundary, just look at the notion of Decision Surfaces in the referenced literature.
[A] https://web.iiit.ac.in/~pjn/M2006/Bayesian.pdf
