Life is a Highway

What is the possibility of extraterrestrial life in the Universe? Mathematical attempts to answer this question have usually wound up the subject of jokes. But I have in good faith made a heartfelt attempt to approach the question from a somewhat different point of view.

To begin with, the oldest stones on the Earth's surface have been dated to 4.5 billion years of age. Moreover, the earliest presence of water on the Earth's surface has been dated to approximately 3.8 billion years ago. Finally, the earliest fossil record, which we can assume marks the beginnings of life upon our planet, dates to approximately 3.5 billion years ago.

This means that it took around 300 million years for life to colonize this planet while it was technically habitable. This number is important because it allows us to estimate the annual likelihood of life forming. Using the average as a metric, we can back-calculate the annual probability of life beginning as follows:

y = 1 - (1 - α)^t { α= annual likelihood of life forming upon planet Earth }
1 - y = (1 - α)^t
ln(1 - y) = t * ln(1 - α)
T(y) = t = ^{ln(1 - y)} / _{ln(1 - α)}

AVG[T(y)] = (¹ / _{(b - a)}) * ∫[a,b]f(y)dy { AVG[T(y)] := average length of time for life to form upon a planet; a := 0; b := 1 }
          = (¹ / _{(b - a)}) * (¹ / _{ln(1 - α)}) * [lim(y)=0→1]((y - 1) * ln(1 - y) - y + C) { C := 0 }
          = -(¹ / _{ln(1 - α)})
       
AVG[T(y)] = ^-1 / _{ln(1 - α)}
ln(1 - α) = ^-1 / _AVG[T(y)]
1 - α = e^{(-1 / _AVG[T(y)])}
α = 1 - e^{(-1 / _AVG[T(y)])}
α = 1 - e^{(-1 / _300MY)}
α = 0.0000000033333333277777777839506172788066
α = 1 / 300000000.50000000027777777777778

So using this planet as a basis, the percentage chance of life occurring upon any planet in the Universe can be estimated as approximately 1 in 300 million.

We can also apply that likelihood to the Milky Way galaxy, which has been estimated to contain approximately 300 million habitable planets, orbiting some 400 billion star systems.

β = 1 - (1 - α)^P { α = annual likelihood of life forming upon a single planet; β = annual likelihood of life forming in the Milky Way; P := number of habitable planets in the Milky Way}
  = 1 - (1 - 3.33e-9)^300M
  = 0.63212055882855767840447622983854
  ~= 63.2121%
 
This estimation is confined to our galaxy alone, which is itself just one amongst billions in the Universe, but you get the idea.