## Saturday, July 18, 2015

### The fallacy of 'billions of billions', or: Why popular arguments that aliens must exist are bogus.

The universe is an awfully big place.  Granted, most of it is empty space.  But within that empty space, there are trillions of stars.  Maybe more.  At least some of those stars have planets around them, and some of those planets are the kind that could give rise to life.  That's still hundreds of billions (at least) of planets that can support life out there.  And so, the popular argument goes, even if the odds of life arising on another planet are very small, there are so many planets that it is bound to happen.  Thus, there almost certainly exist extraterrestrial life forms.  It isn't a matter of if, but of when we find them.

Here's a video of Dr. Carl Sagan presenting a more sophisticated version of this (with actual numbers) to estimate the number of inhabited planets in our galaxy.  Or try this worksheet on the Drake Equation on the BBC website.

It's a common argument.  And it sounds pretty convincing.  If you keep trying over and over, even though something is unlikely, eventually you will succeed.

It's common and convincing, but it's also fallacious.  Here's the problem: How many times do we have to try before we're guaranteed to succeed?

The mathematical answer is infinitely many times.

But that's to guarantee we succeed, with 100% probability.  So a better question might be: what happens to the probability of success as we keep trying?

Let the probability of a success be very low, set to $10^{-X}$, where $X$ is some large number.  This makes $10^{-X}$ a very small number.  Then let the number trials be $10^{Y}$, where $Y$ is some large number.  This makes $10^{Y}$ a very large number.  Now we define a quantity $P_0$, which is the probability of never succeeding, even after $10^Y$ trials.  (If you can't see my math, check your browser's plugin settings)

Assuming whether we succeed or not on a given trial is a simple coin flip with probability $10^{-X}$ of success, then the probability of failure in a single trial is $(1-10^{-X})$.  The probability of never succeeding after $10^{Y}$ trials is just the product
$$P_0 = \left(1-\frac{1}{10^{X}}\right)^{10^Y}.$$

We have said that $X$ is large.  Maybe you remember from algebra learning the formula for continuously compounded interest, where you ended up with an exponential, like so:
$$e^{x} = \lim_{n\rightarrow \infty} \left(1 + \frac{x}{n}\right)^n.$$

Well, in our case, if $X$ is large, then $10^{X}$  is really large, and
$$\left(1 + \frac{-1}{10^{X}}\right)^{10^X} \approx e^{-1} \approx 0.36788$$

If we re-write our expression for $P_0$, then, we find
$$P_0 = \left(1-\frac{1}{10^{X}}\right)^{10^{X + Y-X}} = \left[\left(1+\frac{-1}{10^{X}} \right)^{10^X}\right]^{10^{Y-X}} \approx = \left[e^{-1}\right]^{10^{Y-X}} = e^{-10^{Y-X}}.$$

Now, $e^{-1} \approx 0.36788$ is less than one, so squaring it or tripling it will make it even smaller.  However, taking the square root of it will make it larger.  The resolution comes down to: how does $X$ compare to $Y$?

Consider a simple case, where $X=Y$.  Then $10^{Y-X} = 10^0 = 1.$  So $P_0=e^{-1} = 0.36788.$  That is, there is only about a 37% chance of there being no successes, or in other words, there is a 63% chance of a success happening at some point.  It's not a guarantee, but it's more likely than not.

Now suppose that $Y = X+1$.  This means that we do ten times as many trials as our inverse probability; if the probability is a 1/10, do 100 trials, if the probability is 1/2, do 20 trials, etc.  Then $10^{Y-X} = 10^{1}= 10$, so $P = e^{-10} = 0.0000454$.  That is, the probability of success is 99.995%.  As we increase $Y$, this probability gets even closer to 100%.  Success is all-but guaranteed.

However, now suppose that $Y = X-1$.  This means that we only do a tenth as many as the inverse probability; if the probability is 1/10, do 1 trial.  If the probability is 1/20, do 2 trials, etc.  Then $10^{Y-X} = 10^{-1} = 0.1$, so $P_0 = e^{-0.1} = 0.9048.$  That is, the probability of success is down to a measly 9.516%.  As we increase $X$, this number gets even closer to 0%.

As we can see, our confidence of success depends drastically on the value of $Y-X$.  Even slight differences here can mean huge changes in the probability of success, $P_{\geq1} = 1-P_0$.

 Simple graph showing the steep rise from 0 to 1 in the probability of success.

What this comes down to is whether $X$ is greater than or less than $Y$.  Put differently, does the probability of a single success compare to the number of trials?  Or put in terms of aliens, is the number of planets out there that can give rise to life close to the inverse of the probability of life actually arising?

And the answer is: no one knows!

We do not know how many planets there are.  If we estimate this as $N_p = 10^Y$, then $Y$ might be off by 2 or 3 in either direction.  There might be a thousand times as many as we think now, or there might be only a hundredth of our current guess.  As we just saw, for a fixed $X$, changing $Y$ even by 1 can drastically affect our confidence of extraterrestrial life existing.

Way more crucially, we have no idea how likely it is for life to occur on a planet that can give rise to life.  Think about this.  We have only ever observed life arising on a planet once.  This means that we don't have a very good definition of a planet where life can arise (see above), but it also means that we have a single data point upon which to base a probability.  If I were a pollster, and I went out on the street and asked a single person who they were voting for, and from that concluded that 54\% of voters supported the candidate, you would rightly question my methodology.  If we estimate this probability of life arising on a planet as $p_L = 10^{-X}$, then we don't even know what $X$ is.

Since we do not know what $X$ is, then we don't know what $P_{\geq 1}$ is.  This is a simple model, but it makes its point: Even small differences between $Y$ and $X$ can lead to very different predictions.

Consider again what it would mean for $Y=X-1$.  Take $10^Y$ to be the total number of planets what ever exist or will exist in our universe's lifetime.  Take $10^{-X}$ to be the probability of life ever arising on any given planet in our universe other than our own.  Then if $Y = X-1$, as above, we have $P_{\geq1} \approx$ 10% as the probability of extraterrestrial life ever arising in this universe.  This means that we'd need roughly 10 other universes just like our own before we can be back at roughly 63% probability of life arising again.

The popular statement that the universe is so big that there must be life in it somewhere is a false one.  The universe is quite big, but the probability of life arising can also be so small as to negate this bigness, and we have no way to know if this is the case or not.

The universe can still be as big as it is, and yet still not be big enough for life to arise anywhere else within it.

1. I don't like this

1. You're so sweet.

2. haha you know what I meant, I love you... ;)

2. I like this

1. Thank you! The math on this one was really fun to do. I didn't remember this until later, but I was basically evoking the Law of Rare Events from probability, which says that a binomial distribution will turn into a poisson distribution under certain circumstances.

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6. Hi Reece,

Hey, I'm a writer. I wrote a book called DIY Magic. Check it out on Amazon. I am working on a book right now that has elements of time travel and fantasy. I stumbled on your blog and see that you are a connoisseur of both. I would love to chat with you about some ideas I am working on for a YA novel about time travel. Just to get the opinion of someone who has also thought a lot about it. If interested email me at r.anthonyalvarado@gmail.com

best

7. Hi Reece!

I just discovered your blog, and I just wanted to say how awesome it is to read something by a science/fantasy/scifi enthusiast who is also a Christian. I feel like all of my friends are either into these topics and not believers, or they're Christians but they couldn't care less about them. So thanks!

Also (I haven't read everything you've written yet, so forgive me if you're already addressed this) but I'd love to know your thoughts about how a an actual discovery of alien life would affect your faith and the Christian community as a whole. I realize that you don't think that the probability of alien life is as high as some people think, but it seems to be a popular topic lately so I've been reading and thinking a lot about it--if it would affect the gospel, how it would affect our interpretation of the Bible, etc. I've mostly come to the conclusion that for me, and other reasonable believers, I think my view of God is big enough and mysterious enough that I could expand it to include aliens too. Anyway, just wondering what you think!

1. My position on alien life is fairly agnostic. I don't have any hard and fast proof against alien life, but I have my own reasons for being skeptical. It would take a single alien life form to prove me wrong, but until then I'm going to assume we're alone in the universe.

If we did discover alien life, I'm really not sure what I'd do. I know it would be exciting, but Stephen Hawking's warnings are probably about right. A lot of people are way too overly optimistic about meeting aliens. In real life, if powerful, technologically advanced aliens come to Earth, we can't really get Jeff Goldblum to wire his iBook to the mothership and wipe out their shields; we would really lose that battle, and be either eliminated, enslaved, or farmed. I don't know that aliens ever would attack us (no one does), but it's at least as likely as anything else, and should represent a bigger worry to the people over in SETI.

I wrote this really long and boring post about this, actually. There I was talking about gnomes, but it applies to aliens. I like C.S. Lewis' remarks in "Religion and Rocketry", or the way he explores this in Out of the Silent Planet. I like them because they seem very well-reasoned. However, I'm not sure how to make sense of them in light of the book of Hebrews. I don't expect you to read the post (it's really long and boring), but the point about Hebrews encapsulates what I see as one of the biggest problems to Christianity if we discover aliens.

2. Thanks for the response! Haha I actually didn't think your post was boring at all. That "Religion and Rocketry" essay is really great--never read that before.

I see how you get hung up on Hebrews 2, but maybe it doesn't have to be a deal breaker. I was thinking about that passage, especially how verse 16 says that Jesus came to help Abraham's offspring. The passage is saying that Jesus had to be Abraham's offspring to save Abraham's offspring. Yet, it's not only talking about Abraham's literal offspring that were redeemed--Jesus's death also works to save Gentiles, who are NOT Abraham's offspring (Gal. 3:28-29). Therefore maaaybe it's possible that Jesus's atonement could also be stretched to include other sentient and fallen beings?

Also, in this passage, the point seems to be is that Jesus had to be able to sympathize with humans so that he could be our high priest. These high priest qualifications seem to include that he had to have a physical body and he had to be able to suffer. Therefore if an alien has a physical body and suffering capabilities, maybe Jesus could be their high priest too?

Not sure how I feel about comparing aliens/gnomes to Gentiles. Maybe I'm completely overthinking it. Or maybe I'm just getting sucked into your weird little world.

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8. I have a headache after two hours on your blog and I'm going to sit down and ponder upon it. Thanks for the night of entertainment.

9. You make good points about probability. It is easier to be wowed by a very high number (the number of stars/planets) than to be wowed by a very low number (the odds of life on a planet). In short, rarity can trump multiplicity.

However, I think one thing that favors life on other planets is the window into reality that we are afforded on our own planet. We see on earth how a wide assortment of complex creatures have come into existence from the seemingly bland chemistry of earthly elements. It is more logical to conclude that this creative force (or pattern or phenomena, etc..) is present in the rest of the universe than to conclude that it is absent.

We don't have a 'single data point' with regard to life on earth. We have a vast number of data points covering the vast number of flora and fauna that have come into existence by way of an exceedingly large number of processes and mechanisms spread out over billions of years. The earth has given us a view into this aspect of nature.

We experience this potential of the universe when we walk around the corner at the zoo and are bowled over by the sight of a big, beautiful, living, breathing, wonderfully constructed hippopotamus. Yes, it is true that you are observing this hippopotamus on planet earth- but why should that fact be given so much weight when pondering the prominence of life at any other comparable location? Because the flip side of saying that life may not exist elsewhere is the obvious need to explain why earth would be so special.

In the end, it doesn't boil down to mathematical probabilities. It boils down to whether or not we choose to be provincial in our thinking. It's like saying that the pizzeria down the block is the best pizzeria in the whole world- even though we've never left our hometown.

The universe has shown us the creative power of nature in countless ways and we now know that something like 1,000,000,000,000,000,000,000,000 other planets probably exist. Do we still think our local pizzeria stands out above all the rest?

1. "We see on earth how a wide assortment of complex creatures have come into existence from the seemingly bland chemistry of earthly elements."

Really, we didn't see a wide assortment of complex creatures come into existence from seemingly bland chemistry. Every creature alive right now shares a single common ancestor. So all of those animals at the zoo - elephants, rhinos, giraffes, birds, monkeys - are all just modifications of prokaryotic bacteria. What we have seen is that when there is life on a planet, it evolves and changes to match its environment. But that isn't the same as arising from bland chemistry.

Why is Earth so special? It really isn't, cosmically speaking. It's a totally arbitrary planet, and it just happens to be the one where life emerged. We think it's special because we're from here, but that's anthropocentric thinking. There has to be life on some planet, just because we exist to ponder it; it doesn't matter much which one. If there exists another intelligent species out there in the universe, then I'm sure that they also call their planet whatever means "dirt" in their own language.

It really is a matter of probabilities. If the probability of life (i.e. that first prokaryotic bacterium) arising on a planet is really, really small -- orders of magnitude smaller than the number of planets -- then we can be fairly confident that there is no life anywhere else in the universe. If it is large -- say close to one -- then we can be almost certain that there is life elsewhere in the universe, and lots of it.

If I know that the probability of a pizzeria serving pizza at all is 0.000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,001 (that is, $10^{-66}$), then I don't care if there are 1,000,000,000,000,000,000,000,000 (that is, $10^{24}$, a million million million million) pizzerias out there, the probability of finding pizza outside of my own block is $1-e^{-10^{-42}}$, which is zero to the first million decimal places.

So it isn't provincial thinking, it's cautious and skeptical thinking. I don't know whether the probability for life on a given planet is 1, or if it is arbitrarily close to zero. But I do know that without knowing this probability, the number of planets by itself is insufficient data to predict the probability of life existing elsewhere in the universe.

Since we have insufficient data, the proper answer to the question of alien life is "we won't know until we encounter it."

2. I don't disagree with your mathematical points and I understand the competing probabilities involved. But since you don't know the probability of life arising on a planet your point reduces itself to the math that WOULD APPLY if the probability WERE known. And you’re cautioning people to understand that math.

It's a good essay but your reference to the odds of life on another planet is an intellectual construct. It adds no information to the question at hand since you don’t know what that probability is. You are essentially saying that the odds of life existing on other planets might be low because… the odds of life existing on other planets might be low. Low enough to trump the multitude of planets.

I'm pondering real world factors that shed light on the potential for life elsewhere- leveraging our vast knowledge of how both animate and inanimate systems undergo continual change, transformation, transmutation and growth, with sometimes striking results that we obviously know can lead to life. Supernovas explode with matter and energy, galaxies collide, black holes rip things apart. Elements are forged, radiation emanates, asteroids and comets go hurtling through space sometimes delivering parcels of raw materials, life comes into existence and can evolve and maintain itself for millions of year.

We are given a shining example of this potential in our very lives and on our very planet. It doesn’t matter if you can identify one organism from which all life on earth came. What matters is the grand parade of transformational process that exists in the universe. We’ve been given a front row seat to that process and now we know what the universe is capable of. This is valuable information in pondering what lies outside of our realm.

Returning to one of your other points, you’re assigning a false significance to having 1 planet with life rather than say 632 or 5,456,823 planets with life. There is nothing special about the number 1. It is one number in a massive sea of possibilities. Why not argue that there are 17,398 planets with life? Or 35? It would make just as much sense. Arguing for a number which just so happens to lend special significance to earth reflects an (admittedly understandable) bias. As if choosing between 1 planet having life (earth) vs some other number is a binary choice of two alternatives- when it’s not.

This is mildly reminiscent of the famous ‘Monty Hall’ problem where people think that just because they have a final binary choice between 2 doors it must mean that each door has a 50/50 chance of winning- when that’s not true given the setup. (Google ‘Monty Hall Problem Wikipedia’ if by chance you are not familiar with this.)

The knowledge of 1 planet with life would only be special in one circumstance. And that is if zero planets with life were known at the time (not even earth) because this discovery would confirm that planets with life can indeed exist. But since we already know that at least one planet with life exists (earth) then that breakthrough knowledge has already occurred.

Taking a step back, I realize you are not saying that earth is the only planet with life. You are saying that we just don’t know. But part of pondering the question is to see what it’s like to make an argument for only earth having life. I see it as unlikely given what I’ve observed and learned and how big of a laboratory the universe is. By the way my name is John. Cheers.

3. We are all part of the exact same "life" event... As far as Im concerned, life occurred once. We all share the same DNA processes that are very likely unique to earth life. We have a single example of life spontaneously forming. We dont know how likely that is. Just because that life survived and became diverse doesn't mean much in terms of extraterrestrial life.

The real question is... how many times did life occur on earth independently? Was there some other prokaryote using a different DNA simile? Were there a million different ones and only one survived?

Life happened on earth once in certain conditions, but we have no evidence of it happening more than one single time. That's the part that gets me, and that's probably the most relevant part to the topic at hand. Even when you KNOW conditions can allow for living organisms, it's STILL exceedingly rare.

10. have noticed the effects happening To the bible. But none in old testament? Let me know of any if you have examples.

he jfk footage was smoking gun proof to me. All.museums movies, reenactments thru out the decades use a 4 seater car.
I would have dismissed the stein bears thing. But in my timeline it was a 4 seater car. Not 3 and only 4 people. Jackie only female. Black and white.
The Volvo logo is diffeeent also, with n history of it not ever having the arrow at the top.right. I’ve heard these quantum effects have been happening since the 50s.
It’s a little ironic that cern posted on social media on Oct 21 2015 the back to the future 2 screening this is also the date mcfly lands in the future. What does cern have invested on this movie to promote it. Cern is a Hadron collider so why are they promoting a movie from the 80s.

There's a new ritual at cern which is very pagan like, where they are dancing around with a clock behind them spinning in multi directions. Symbolism. ..
Quantum computing exists as of now. This is all very recent

Http:// granitespraybeaumont.com

11. I was very sceptical, when I started reading this post, but that math you showed is convinicing and accurate. The point you're making is very intuitive too, since you're basically just saying, that the number of events is useless without the probability of a positive event and since X and Y are powers of ten, it makes sense, that you would get a logistic function.
This way, you could get the 10% in case Y = X - 1 just by calculating the expected value of 10^-X*10^Y.

However, I think that scientists like Carl Sagan, who consider life to be plenty in the universe, just assume, that the probability of life in the universe is, in terms of your X and Y, much larger than the number of planets. Of course, no one knows for sure how high that probability is, but we can at least make an educated guess. This way the predictions will of of course wary between "we are alone" and "there is life everywhere", which is why I think the arguments for or against those positions are much more interesting, than a quantitive probability.

I was in the "alien life"-camp, but your article showed me, that I need to inform myself better on the conditions for life to be able to start guessing on the probability of life.
Kudos to you, for reminding us, that rareness is able to trump plentyness of events and that just saying there are many planets is not a satisfying argument.

12. What a very interesting post. Thank you.