Unlikely Events Can Happen If There Are Enough Trials

Suppose an event has one in a trillion (10^-12) chance of happening. How many trials are needed for it to happen? We start with the following expression

(1-p)^1/p

which is the probability of the event not happening in 1/p trials. It has been shown that as p tends to zero the value of the above expression tends to 1/e where e is the base of the natural logarithms (about 2.72) so the value of the limit is about 0.37. (For p equal to 0.2 the value of the expression is 0.8⁵ or about 0.33 so the limit is approached for even relatively large values of p.) For p = 10^-12 the probability of the event not happening after a trillion trials is 0.37. For 100 trillion trials the probability of the event not happening is 0.37¹⁰⁰ or 6^-44. Therefore the event is very likely to happen in 100 trillion trials.

There are 100 billion galaxies in the universe, each having over a billion stars. Therefore the number of stars exceeds 10²⁰. It has been argued that life cannot exist elsewhere in the universe given the very small probability of it happening on earth. The above analysis shows that even if such probability is only 10^-18, the sheer size of the universe suggests that it is highly likely to happen in other planets besides the earth. Of course such planets maybe many light years away, so we may never know about them.