I very strongly believe that aliens exist, especially the microbial type, but also the intelligent type. Just look at the [very large] numbers and you'll know what I mean. I use exponential form as it makes working with large numbers much easier.
Number of stars in an average galaxy: 1E11 (100 billion)
Number galaxies in the universe: 2E11 (200 billion)
Number of planets and fair-sized moons around a star on average: 2.5E01 (25)
Number of planets and fair-sized moons in the universe: 1.E11*2E11*2.5E01, (1*2*2.5)E(11+11+1), 5E23 (500 sextillion (aka 500 billion trillion))
That's for starters. There are conditions though:
The galaxy must be "new" enough for metals (elements beyond helium) to form. You can't have life without carbon, hydrogen, oxygen, and nitrogen (CHON as I remember it). This eliminates about 1 in 25 possibilities.
The galaxy must be stable. Colliding galaxies cause so much mayhem from things like star burst and the rare collision that life will be extremely difficult to develop even if other conditions are suitable. This removes about 1 in 5 galaxies.
Stars must fall within the galactic habitable zone. Too close to the galaxy center, it's dangerous due to stars more likely going supernova inhibiting. Too far out and metals are lacking. This is about 1 in every 10,000 stars as a rough estimate.
The star must be of the right type. There are 7 main star types (classes), in order from coolest to warmest: M, K, G, F, A, B, and O. Our own star is G2, near the upper (warmer) edge of the G-glass. The A, B, O, and upper 1/3 of the F classes have too short of a duration for intelligent life to develop but they are fairly rare. The M class and lower 1/3 end of the K class live long, but the planet risks being tidally locked (like our Moon is to Earth), but are very common. This, thus, covers about 1 in 20 stars (for intelligent life, 1 in 2 for microbial).
The planet or moon must fall within the habitable zone of a star. Too close to the star and the planet scorches. Too far and the planet is frozen. Even something as far out as 12 AU (between Saturn and Uranus; based on our Sun) could still have life, but must be tucked far under the surface of a planet or moon). Close-in planets could still be okay, mainly if tidally locked. This eliminates about half of the possibilities.
The star must be stable. A young star is quite unstable and fierceful. An old star is also unstable, from running out of fuel. Thus, the star needs to be of the right age. This eliminates about 1 in 20 stars.
The planet or moon must have a reasonable, fairly circular orbit. Planets with wild elliptical orbits will have short bursts of great heat followed by a long period of great cold making for an unstable environment. Many planets seem to have elliptical orbits. This eliminates about 1 in 10 planets I'd say.
The planet or moon must have some form of protection against hazards such as solar flares - an ocean, atmosphere, deep caves, and/or magnetic field of sufficient strength. This eliminates about 1 in 5 planets.
The gravity of the planet or moon must be just right. Too small of a planet means too little gravity to hold on to an atmosphere. Too large of a planet and getting around will be extremely difficult. This removes 1 in 10 planets.
Techtonic activity is necessary for recycling old material. Mars is practically dead and there hasn't been any known techtonic activity for billions of years. I would say that this eliminates 1 in 5 planets.
Finally, the planet or moon must have a solid surface not under an excessively thick atmosphere. Sure Jupiter or Neptune may have something solid far below, but this is going way too far. Venus is pushing it. This eliminates about 1 in 50 planets.
So, let's sum up the result:
5E23 / 2.5E01 / 5E00 / 1E04 / 2E01 / 2E00 / 2E01 / 1E01 / 5E00 / 1E01 / 5E00 / 5E01 =
(5/2.5/5/1/2/2/2/1/5/1/5/5)E(23-1-0-4-1-0-1-1-0-1-0-1) = 0.0004E13 = 4E09
That still 4 billion possibilities in the universe for intelligent life. I would still say that there's a few hundred in our galaxy alone (us being one of those)!