The Probability That You Were Born

Why was I born when the odds of such an occurrence are infinitesimal? Specifically me and not some other person in my place. Let’s see just how unlikely this is. Only one sperm will combine with the egg to form a zygote, so the sperm that made you had a 1/280,000,000 chance of making it (280 million sperm in one ejaculation). That’s per ejaculation. If we consider how many non-procreative ejaculations the average human male undergoes in one lifetime the odds get much lower. Then consider that only 1/7,000,000,000 males were your dad and 1/7,000,000,000 females were your mom. The odds of them choosing each other to mate with are 1/4.9*10^37. Remember these are the odds that you are born, not someone else, but you. We are already at the order of 1 in 10^49 if we assume the average male ejaculates once per day over the average life span of 75 years (yes I know, I am laughing by this point as well). So why was I born then? The answer could be that the odds of anyone else being born were equally low, so why not me. But that isn’t quite right because to calculate the odds of anyone except me being born we add together the odds of all 10^49 possible different people besides me being born to get a number of the order 1 in 1. So it seems like the overwhelming probability is that someone who isn’t me should have been born instead of me. Now if we really want to look deeper, we must multiply the odds of our mother being born with the probability of our: father, grandfather, grandmother, great grandmother, great grandfather, etc. This is because all of these people being born are necessary conditions to your birth (if your grand-dad doesn’t exist you don’t). All of these numbers are of the order 10^49, so the odds of me being born are really 1 in 10^(2n*49) where n is how many generations my lineage goes back. So now we have a truly massive number. At this point the odds of me being born are much less than those for entropy to reverse. That is, it is more likely for certain phenomena to happen backwards in time rather than forwards in time. It is more probable that when you pour milk into your coffee, the milk will jump back up into the container instead of mixing uniformly with the coffee, than you being born. We could continue calculating by considering the probability of life occurring at all much less on this specific planet. At this point the odds are closer to infinity than any finite number you can think of so our error would be lower considering the odds to be 1/infinity, which is equivalent to zero (in the limit, which is as equivalent as any infinity gets to anything else). But I don’t want to obfuscate a discussion of how likely your birth is with how likely life is. I’m trying to make a point about how large numbers and probabilities mesh with our intuitions.
This prologue raises some questions about how seriously we should really take probabilities. There are a few ways out. The first is that when we live in a universe as old (13.6 billion years) and as vast (the observable universe is 93 billion light years wide) as ours, there are so many events total that any probability will involve very large numbers. That is 1 in order 10^49 isn’t very improbable when we consider that the number of all events possible throughout all time and space is closer to infinity that to any finite number we can imagine. So here we can use relative probabilities and see that our birth, while improbable in absolute terms, is still much more probable than any of the other events the universe can and has been witness to just by the sheer number of things that can happen. Though here I must note something funny about probabilities and massive numbers that approach infinity. If we have an infinity of balls and we pick out one, what are the odds of getting the ball we got (let’s say the red ball). Well that’s easy, the odds are 1/infinity, so….zero (in the limit, in other words the answer is destined to get closer and closer to zero as we increase the accuracy so it must equal zero). The odds of getting the ball we just picked out is ZERO. This is even more bizarre when we consider that the odds of picking any given ball out of the infinite pool is also zero, and yet we know for sure that we will get one of the balls when we pick. That is we know for sure that an impossible event will occur. Now I can get even more weird here and ask the probability of picking 36,000 red balls in a row. That probability is (1/infinity)*(1/infinity)…. 36,000 times. That is 1/(infinity)^36,0000 which is still equal to 1/infinity (just trust me here, if you really want to see the math post in the comments and I’ll supply a proof and calculation). So the odds of picking one red ball out of the pool is the same odds as picking any given finite number of red balls in a row out of the pool, as long as the pool is infinite. I believe this is a paradox in probabilities, though feel free to correct me, I’d be quite intrigued to see how I am wrong here. I’d be very impressed if someone understands limits well enough to figure out the picture here. Here’s a real paradox: How many sizes of infinity are there? For those well trained in mathematics, you will know that there are in fact an infinite number of sizes of infinity. By taking the power set of any given infinity you make a larger infinity. However, what size of infinity is the number of different sizes of infinities? I’d be much more impressed if anyone proves me wrong here because I’m certain I’m right about this. Good luck with those. Don’t worry, tomorrow will be lighter reading.

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