First, let me define my meaning of "alien". An alien to me could be any single celled microorganism that is either unknown in science or unknown of this planet. So with that being said , I have to weigh the possibilities of such "creature" or "alien" to be of existence in the known or observable universe.
With the scientific facts that are available to us now , such as the vastness of the universe and the amount of galaxies within , math would be the arguable factor in an answer to the question "Do aliens exist?".
Facts to consider when thinking of an educated answer to this question ...
1. The Milky way in which our solar system located ,is estimated to contain 200 - 400 billion stars. Recent estimates give the galaxy a population of at least 50 billion planets, 10 billion of which could be located in the habitable zone of their parent star. In 2011, new evidence obtained via gravitational microlensing indicated an average of at least one bound planet per star in the Milky Way, with Earth-sized planets being more numerous than gas giants.
We may also want to consider that this can be compared to the one trillion stars of the neighbouring Andromeda Galaxy.
2. The "known" and or "The observable" universe contains about 3 to 100 × 1022 stars (30 sextillion to a septillion stars), organized in more than 80 billion galaxies, which themselves form clusters and superclusters.
With these facts considered , I find myself having trouble not to believe in "alien life " somewhere within that many planets. I would be a fool to think that "we" as the humans we are , be the only living thing in this universe in which we live.
I believe in aliens !