Drake's Equation
(slightly modified) Drake's equation for SETI

N = R n_e f_l f_i f_c * L

N is the number of civilizations in our galaxy* we can detect communications from.

R is the rate of star formation, which is about 10 per year per large galaxy.

n_e is the average number of planets per star that can support Life.

f_l is the proportion of potentially Life-supporting planets which actually develop Life.

f_i is the proportion of Life-supporting planets which get intelligent Life.

f_c is the proportion of civilizations which develop technology which release detectable signals into space.

L is the duration over which such signals get released.

Why does Drake's equation just look at our galaxy? Signals could be coming from much further away; OTOH we can't get (in person) anywhere near the far end of our galaxy. So we'll consider (a) our galaxy, (b) most of the observable universe, and (c) a small space that we can actually travel to.

Things that have changed in typical estimates of this:
* estimates of the size of the observable universe keep going up
* estimates of the average number of planets per star keep going up
* estimates of how fast life evolved on Earth (and therefore how likely it was to evolve) keep going up

Making the estimate the normal way first:

number of civilizations that should be sending us signals, PER GALAXY, assuming civilizations only develop on planets

Just for us, let's add a factor: split L into how long we'll last if we blow ourselves up, and how long we'll last if we don't, and take a weighted average.

10 10 0.1 0.000001 0.1 (1000 99.99% + 10^10 * 0.01%)
= 10 10 0.1 0.000001 0.1 * (100000)
= 10^1 10^1 10^-1 10^-6 10^-1 * 10^5
= 10^-2

number of civilizations that should be sending us signals, PER GALAXY, without assuming civilizations only develop on planets

At first sight you might think this doesn't go up by all that much if you include non-planet environments, because we estimate the chances of life becoming intelligent enough to send signals as very small.

But on the other hand, some places might host life that evolves much much much faster than Earth life (specifically, maybe neutron stars).

We estimate that 10^-5 is the proportion of civilizations that are on neutron stars, and that life on neutron stars develops 10^7 times as fast. So this puts our overall number up to:

10^-2 + (10^-2 10^-5 10^7)
= 1

number of civilizations that should be sending us signals, WITHIN VISITING DISTANCE

We think we would be bothered to visit any civilization within 1000 years' travel. Call this 1000 light years, because once we've got a spaceship up to near the speed of light it's going to stay at that speed easily. This means we're looking at a volume of 10^-6 of the galaxy, but because the galaxy is flat that's actually 10^-4 of all the stars in the galaxy.

So our estimate for this heading is

10^-4

number of civilizations that should be sending us signals, at all

Say out to half the radius of the observable universe (because further out than that we might be seeing civs before they've had time to develop enough). So that's about 1/10 of the galaxies in the observable universe.

10^10

This is the main number in a way, because this shows why we've got the Fermi Paradox.

number of civilizations that should be sending signals, even if they're too far away for us to receive

10^11

whole universe, under some people's assumptions about the Big Bang producing a finite universe

10^11 * 10^300 = 10^311

10^11 * ∞ = ∞

SETD (Search for Extraterrestrial Dumbness)

The amount of life elsewhere can really easily be worked out from the numbers we've already got.

number of biospheres that should exist, PER GALAXY, assuming life only develops on planets

= 10^1 10^1 10^-1 * 10^9

= 10^8

= 10^10

number of biospheres that should exist, WITHIN VISITING DISTANCE

10^4

I think this is another important number.

10^19