Despite the rational expectation that your vote will not turn an election, you may, like many of us, vote anyway, and for various reasons. Some people are proud of the effort they put into being informed and involved. Others vote out of a sense of duty or because of their belief in the value of democratic participation.
Though it takes time and effort to learn the issues as well as the positions and backgrounds of the candidates, this effort may boost your pride and sense of belonging in the community. Just as the pursuit of knowledge in the classroom can expand your horizons, the struggle to understand and shape the politics of the day can profoundly increase your sense of control over the issues that affect you most. Like other citizens, as you become more interested in politics, you become more likely to vote and to participate in other ways.
Many people's decisions to vote are also shaped by a sense of duty or obligation. They - like you - may understand how unlikely it is that each individual vote will make a difference. But they - like you - may also believe that there is something inherently good for the community and for themselves in participating in elections. Standing up and being counted - showing that you care about what happens in the political system - is one of the benefits of voting.
Voters may also recognize a conundrum in the calculation of people who believe that "it's not logical to vote, because my vote probably won't make a difference." In the limit, if everyone decides not to vote because no one believes an election will be close enough to be decided by any one person's vote, then one vote will decide the election. Similarly, if many of those who share your political preferences share also this belief and abstain from voting, the candidates and policies you and they least prefer would likely be victorious. Widespread belief that "my vote won't make a difference" leads logically to circumstances in which one vote may make all the difference.
Finally, many people also vote in the belief that broad and sustained participation is critical for the legitimacy of a democratic system. They recognize that elections would not be democratic without their own and others' participation.