Given the voting history of an individual (voted or not for each of the past $n$ elections) calculate the probability that the individual will vote in the upcoming election.
My first thought was simply to calculate the proportion of times voted to however many elections there have been, but now I'm not so sure because on second thought the decision to vote is not an independent event. Rather, it is influenced by one's previous voting habits (for example, those who have voted in the past will keep voting).
Is there a name for what I am trying to compute? Is it possible to compute? If so, how do I compute it?
These kinds of problems generally require you to make certain assumptions. As you remarked correctly, there are countless of aspects that can effect someone's decision to vote.
However, in simple homework questions, the professor will generally want you to make the kinds of assumptions that will lead you to an answer (this is just the state of education atm). So here the "correct" assumption is that the frequency of voting (the number of times the person voted divided by the number of elections since he could start voting) is representative of the probability of voting in the next election.
If you don't have any other information, this is arguably the best kind of guess you can make.
The general problem of coming up for probabilities of future events, given past data is called "statistical inference", and it is quite cool.