Help me understand Bayesian prior and posterior distributions 
In a group of students, there are 2 out of 18 that are left-handed. Find the posterior distribution of left-handed students in the population assuming uninformative prior. Summarize the results. According to the literature 5-20% of people are left-handed. Take this information into account in your prior and calculate new posterior. 

I know the beta distribution should be used here. First, with $\alpha$ and $\beta$ values as 1? The equation I found in the material for posterior is 
$$\pi(r \vert Y ) \propto r^{(Y +−1)} \times (1 − r)^{(N−Y +−1)} \\
$$
$Y=2$, $N=18$
Why is that $r$ in the equation? ($r$ denoting the proportion of left-handed people). It is unknown, so how can it be in this equation? To me it seems ridiculous to calculate $r$ given $Y$ and use that $r$ in the equation giving $r$. Well, with the sample $r=2/18$ the result was $0,0019$. The $f$ should I deduce from that?
The equation giving an expected value of $R$ given known $Y$ and $N$ worked better and gave me $0,15$ which sounds about right. The equation being $E(r | X, N, α, β) = (α + X)/(α + β + N)$ with value $1$ assigned to $α$ and $β$. What values should I give $α$ and $β$ to take into account prior information?
Some tips would be much appreciated. A general lecture on prior and posterior distributions wouldn't hurt either (I have vague understanding what they are but only vague) Also bear in mind I'm not very advanced statistician (actually I'm a political scientist by my main trade) so advanced mathematics will probably fly over my head. 
 A: A beta distribution with $\alpha$ = 1 and $\beta$ = 1 is the same as a uniform distribution.  So it is in fact, uniformative.  You're trying to find information about a parameter of a distribution (in this case, percentage of left handed people in a group of people).  Bayes formula states:
$P(r|Y_{1,...,n})$ = $\frac{P(Y_{1,...,n}|r)*P(r)}{\int P(Y_{1,...,n}|\theta)*P(r)}$
which you pointed out is proportional to:
$P(r|Y_{1,...,n})$ $\propto$ $(Y_{1,...,n}|r)*P(r)$
So basically you're starting with your prior belief of the proportion of left handers in the group(P(r), which you're using a uniform dist for), then considering the data which you collect to inform your prior(a binomial in this case.  either you're right or left handed, so $P(Y_{1,...,n}|r)$).  A binomial distribution has a beta conjugate prior, which means that the posterior distribution $P(r|Y_{1,...n})$, the distribution of the paramter after considering the data is in the same family as the prior.  r here is not unknown in the end. (and frankly it wasn't before collecting the data.  we've got a pretty good idea of the proportion of left handers in society.) You've got both the prior distribution (your assumption of r) and you've collected data and put the two together.  The posterior is your new assumption of the distribution of left handers after considering the data. So you take the likelihood of the data, and multiply it by a uniform.  The expected value of a beta distribution (which is what the poster is) is $\frac{\alpha}{\alpha+\beta}$.  So when you started, your assumption with $\alpha$=1 and $\beta$=1 was that the proportion of left handers in the world was $\frac{1}{2}$.  Now you've collected data that has 2 lefties out of 18.  You've calculated a posterior. (still a beta) Your $\alpha$ and $\beta$ values are now different, changing your idea of the proportion of lefties vs. righties.  how has it changed?
A: In the first part of your question it asks you to define a suitable prior for "r". With the binomial data in hand it would be wise to choose a beta distribution. Because then the posterior will be a beta. The Uniform ditribution being a special case of beta, you can choose prior for "r" the Uniform disribution allowing every possible value of "r" to be equally probable.
In the second part you have provided with the information regarding the prior distribution "r".
With this in hand @COOLSerdash's answer will give you the proper directions.
Thank you for posting this question and COOLSerdash for providing a proper answer.
