Given a scatterplot, is it possible to determine the probability that a new data point will land in a certain region of the graph? I was asked to determine the probability of a new data point landing in a certain rectangular region on the graph given other data points that make up the scatterplot. If that is not possible, is it possible to determine the probability the dependent variable falls in a certain range given the independent variable? 
Thanks
 A: For your first question, you can do # of points in rectangle / # of points in scatterplot. That should give you an unbiased estimate for the probability of a new one falling in the rectangle (you're doing a monte-carlo, essentially).
For the second question, I assume you have the independent var and want to know if the dependent one will put you in the square. You need to fit a model to derive a-priori distribution for the dependent variable, given the independent one. You can then use this distribution to derive the probability you need.
A: It seems like you really just want to estimate the distribution function of the data.
A rectangle on the graph just corresponds to a particular set of bounds on $X$ and $Y$. In general, the probability that point $P$ "falls in a  certain rectangular region" $R$, with boundaries $x_0, x_1, y_0, y_1$, can be expressed as
$$
F(x_1, y_1) - F(x_0, y_1) - F(x_1, y_0) + F(x_0, y_0)
$$
where $F$ is the joint distribution function of $X$ and $Y$. There's a nice and easy derivation in these lecture slides.
You can estimate empirical bivariate CDFs directly. There used to be an R package for this, but it has been removed from CRAN. You can get the source code, as well as a helpful vignette, from the CRAN archive or the CRAN GitHub mirror.
You can also do Monte Carlo, as mentioned in the other answer. Or if you have a bivariate parametric model in mind, you can use that to derive your probabilities as well.
The second problem can be approached differently. You just need $F(y_1\ |\ x_0 \leq X < x_1) - F(y_0\ |\ x_0 \leq X < x_1)$. Many statistical models are based on assumptions about $f(y\ |\ x=X)$, from which you can derive an estimate of $F(y\ |\ x=X)$. Just about anything estimated with maximum likelihood would work here, as would a Bayesian model.  In fact, in a Bayesian franework this calculation is standard, and is referred to as the "posterior predictive distribution".
