House of probability: a puzzle

This post will start with a puzzle of sorts which is not supposed to be difficult but rather thought-provoking. Enjoy!

Room 1

You are led into a room with 5 bins. Inside each bin are playpen balls. Each bin contains a single ball color – red, orange, yellow, green, or blue. You are instructed to take a black bag and place inside it exactly 200 red, 30 orange, 25 yellow, 20 green, and 2 blue balls. You follow through with these instructions. Then, you are instructed to stir and mix up the balls. Finally, you are instructed to reach in blindly and pick a single ball at random.

What color ball are you most likely to pick? What color ball are you least likely to pick?

Room 2

You are now led into a second room through a door. There you find a tied-up black bag. You are given the following information:

  • The black bag contains playpen balls.
  • There are multiple ball colors.
  • There are differing amounts of each ball color.
  • The balls have been mixed up.
  • (In other words, it was premade in a similar manner to room 1 but may have different colors and amounts).

You are then instructed to reach in blindly and pick a single ball at random. You come up with a purple ball.

Is the purple ball likely to be the most common color? Is it likely to be the rarest color?

Room 3

You are now led into a third room through a door. There you find a tied-up black bag. You are given the following information:

  • The black bag contains playpen balls.
  • There are multiple ball colors.
  • There are differing amounts of each ball color.
  • The balls have been mixed up.
  • (In other words, it was premade in a similar manner to room 1 but may have different colors and amounts).

You are instructed to empty the bag of balls on the floor and pick your favorite color. This happens to be seafoam green.

Is the seafoam green ball, your very favorite color, likely to be the most common color? Is it likely to be the rarest color?

Room 4

You are now led into a fourth room through a door. There you find a tied-up black bag. Also, surprisingly, sitting on the floor in a golden playpen ball. You are given the following information:

  • The black bag contains playpen balls.
  • There are multiple ball colors.
  • There are differing amounts of each ball color.
  • The balls have been mixed up.
  • (In other words, it was premade in a similar manner to room 1 but may have different colors and amounts).
  • Also, the golden ball came from inside the bag.

Is the golden ball likely to be the most common color? Is it likely to be the rarest color?

Room 5

You are now led into a fifth room through a long, dark hallway. Please, sit down in this comfortable chair. This is where everything will be explained to you. In room 1 you know the most common and rarest color in the bag because you actually counted out the balls and placed them in the bag. Therefore, your random selection will likely be the most common color, red. In room 2, however, the bag of balls was premade. Since you drew at random, you would expect statistically that whatever you draw will be the most common color. Therefore, you can infer that the purple is the most common color in this bag. In room 3 you picked your favorite color. Well, this is not random! Therefore, you cannot make any sort of inference about how common seafoam green balls are in this bag. Finally, in room 4 the bag is premade and a golden ball has already been selected. Herein lies the problem to answering the questions posed. You do not know how the golden ball was selected, whether randomly or someone’s favorite color or some other method.

All of these lessons show us that in order to make the statistical inference that the sample is likely to be a common type, it must be known to be a random sample. This is absolutely central. What is this exercise relevant to? Well, the Earth is our sampled life-producing ball, and the universe is a black bag full of other balls. Is the Earth a random sample? The insurmountable problem is that from our perspective we simply do not know if the Earth is a random sample. It’s like we walked through the door (i.e., became conscious as a species during evolution) and there is a golden ball and we look up at the starry sky which is a black bag full of unknown colors. Our situation is room 4. We cannot say whether the life-producing planets are common or rare because we do not know if we are a random sample or not.

What is this arguing against? Ultimately, it is arguing against the Principle of Mediocrity (PoM). More specifically, a version of the PoM which is statistical in nature, which IMO is the only PoM that really matters. The PoM states that humans represent a random sample, therefore are likely to be common. The problem with this is that we do not know if we are a random sample. We did not draw humans out of a black bag randomly, so how can we possibly know if we are a random sample? It’s not enough to just assert that we are a random sample, rather we need to prove this. There needs to be reasons and evidence. Just like if you walked into room 4, you would have to prove that the golden ball was randomly selected before you could make the statistical inference.

Some reasons that might be given are the discovery of earthlike exoplanets using the Kepler Space Telescope. However, the Rare Earth Hypothesis pushes against this by saying that the number of coincidences required to evolve from abiogenesis to intelligent species are statistically improbable despite earthlike planets being relatively common.

There is another version of the PoM that seems to be the more common one on the internets. This one irreverently states that humans are mediocre chemical scum on a piece of dirt in a meaningless sea of universes in an abyss of nothingness that came from nothingness. This version, which I have caricatured here, I like to call the value-based PoM. It’s aimed at saying that the physical configuration of humans and earth is basically no more or less valuable than any other physical configuration. Its proponents probably don’t even realize that they are preaching amoralism. There’s a reason why I don’t hesitate to push my lawn mower over crawling insects yet I would never push my lawn mower over crawling infants. I value little humans more than mosquitoes. You can hammer a nail into wood but you would never hammer a nail into your spouse. I don’t think they would disagree, so I think their statements stem from a misunderstanding of the statistical PoM. I think they really believe the statistical PoM without realizing that they are merely asserting we are a random sample without justification and then apply our alleged commonality to the idea of value thereby allowing them to formulate cheeky, controversial statements which are metaphysically loaded and contradictory to secular humanism and all other humanisms. /rant over

Another argument I found for the PoM is put forth by famous cosmologist, Alexander Vilenkin:

Actually, I am surprised that this issue is so controversial, since one can easily convince oneself that the Principle of Mediocrity provides a winning betting strategy. (1)

He goes on to give an example: You show up to a scientific meeting in which everyone is wearing colored hats and there are no mirrors in the room so you do not know what color hat you are wearing. You count 80% white hats and 20% black hats. If you have to bet on what color your hat is, you should bet on white, because you should assume that you are randomly selected.

Vilenkin is correct! But, his scenario does not reflect our situation at all. His scenario is closest to room 1 in this blog post puzzle. If I were to make some corrections to his analogy, it would look like the following. You show up to a scientific meeting and are blindfolded. A hat is placed on your head. You are told that everyone else has hats as well. You are told that your hat is seafoam green. You are still blindfolded and asked if you think anyone else has a seafoam green hat. Should you assume you are a random sample? In this more accurate scenario, there is no winning betting strategy precisely because of a lack of appropriate information. For example, if the hatter had told you that your hat was randomly selected, then you could make the inference. Knowing that what we are looking for is a random sample it crucial to making the statistical inference of the PoM, and this information is simply not available to us.

In conclusion, if we are asked if humans are mediocre and common OR exceptional and rare, I think the response is that we don’t know. Based on scientific data we still don’t know. Based on statistical inferences we still don’t know. This means to get anywhere we should start with new science. Only with new discoveries can we get closer to knowing how common or rare Earth actually is.

(1) “Principle of Mediocrity” by Alexander Vilenkin, published on Arxiv 2011.

We are alone in the universe (part 2)

You are probably wondering what evidence one could possibly be brought to the table that would suggest that we are alone. If you are wondering this, you are in the right place. Get ready to rumble.

From part 1 we learned that the Principle of Mediocrity suggests that the frequency of complex earth-like life is an indicator for the frequency of intelligent life regardless of how exotic extraterrestrial life ends up being. This is because humans are much more likely to be an easy way to evolve intelligent life rather than a difficult way. We are not special like a snowflake; we are mediocre. But, that’s OK. Actually, that’s a good thing for this analysis.

The central question now is: what do we think is the frequency of complex earth-like life? The general feeling of space enthusiasts is that our universe teeming with microbes and with intelligent civilizations popping up a few times per galaxy or so. Feelings and guesses are fine and dandy, but there is a more reasoned approach that has concluded that earth-like complex life is incredibly rare. This is the Rare Earth Hypothesis. The analysis goes like this: we can see to produce complex life and intelligent life on Earth, several factors were important including:

  1. Galactic habitable zone – not too close to the central black holes which emit gamma radiation, not too dense region of stars which poses danger of supernova and gravitational perturbations
  2. Favorable star – must have adequate lifespan for evolution
  3. Planet in Goldilocks zone – allows for liquid water
  4. Good Jupiter – protects from asteroid impacts (Bad Jupiter refers to a gas giant in a closer orbit to the sun than the earth and would cause detrimental gravitation perturbations)
  5. Stable orbit – for climate stability
  6. Planet composition – need solid surface in addition to oceans of water
  7. Plate tectonics – for carbon cycling and greenhouse effect
  8. Magnetosphere – protects from harmful radiation
  9. Billions of years of stable climate – don’t freeze or have runaway greenhouse effect, just look at Mars and Venus to see what could have happened to Earth
  10. Abiogenesis (or panspermia?)
  11. Abiogenesis occurs early in planetary life
  12. Not too many mass extinctions
  13. Other factors

Any individual factor is not likely rare in itself (except the factors which must stay true over long time periods). For example, we know extrasolar planets are not rare. Statistically every star has at least one planet. Also, planets in the Goldilocks zone are not rare. According to Kepler Space Telescope data around 20% of stars have rocky planets in the Goldilocks zone.

The lesson we learn here is that it’s not that individual factors are rare, it’s that the combination of factors is rare. It’s like rolling a cosmic dice over and over and having to get the right combination of numbers by chance. Chance and coincidence are at work here. There are 12 factors listed above, but how many factors are there really? There could be far more, but we don’t know for sure. Furthermore, we don’t know the frequency of each of these factors yet.

Let’s perform some calculations to get a feel for how chance will affect the frequency of complex earth-like life. Looking out at the observable universe there are about 100 billion galaxies each with about 100 billion stars. That means there are 10^22 stars in the observable universe! Now, let’s make some assumptions for the sake of analysis. Let’s assume there is one planet per star. Let’s further assume there are 22 individual planetary factors (like the 12 listed above) necessary for complex life and each factor has a 10% chance of occurring. How many planets will harbor complex earth-like life? With these assumptions, only one single planet in the whole observable universe will! What if we increase the average frequencies of the factors to 20%? There will be just over 4 million planets with complex earth-like life which is far less than one per galaxy. What if we increase the number of planetary factors to 400, what would the average frequency need to be for just 2 planets with intelligent life? About 88%. Doing these calculations is constrained by our starting assumptions, but this exercise is helpful because it shows us how the universal lottery may require substantial luck just for a few planets in the observable universe with complex earth-like life.

The thing that pushed me over the edge in this discussion is the factors which must remain true over very long time periods. Complex life is very fragile and that is evidenced by the extinction of so many species. How many dinosaurs have you seen today? If you go to the Creation Museum then Adam and Eve walked alongside dinosaurs, but the fossil record completely fails to support this. The dinosaurs were wiped out during a mass extinction event around 65 million years ago partly caused by a 10 km diameter asteroid slamming into the Earth causing severe climate change. About 50,000 years ago there occurred a similar event called the Toba catastrophe theory which nearly wiped out all of humanity. It is thought that the human population was reduced to around 6,000-10,000 individuals! How lucky are we to have persisted? These kinds of extinction events are common in the fossil record, and even more interesting may help accelerate evolution by opening up niches. Evolving complex life may require a delicate balance of extinction and speciation. But, how often does a delicate balance happen by chance in the universe?

Astrobiologist, David Waltham thinks that the most lucky feature of our planet is its 4 billion years of climate stability. Think about our neighboring planets who probably started out with compositions similar to that of Earth. Due to the sun’s gradual increasing solar output and a runaway greenhouse effect, the surface of Venus is more than 400 C, far too hot for earthly life of any sort. Even extremophiles would find this to be hell. And, Mars once had oceans of water and possibly life, but now is a freezing desert and bombarded with lethal doses of radiation. It may have pockets of microbial life, but certainly nothing complex like on Earth. We are lucky to have enjoyed such climactic stability.

How often does abiogenesis occur? How often do earth-like planets fail to produce complex life from simply life? How often on earth-like planets does extinction events set back evolutionary progress? How often does a planet enjoy 4 billion years of climate stability? If your answer is, “Not very often” then you might be a proponent of the Rare Earth Hypothesis.

We are alone in the universe (part 1)

Suppose there are many different ways for the universe or multiverse to evolve intelligent life. There will be easy ways to evolve intelligence and difficult ways, and these ways will fall on a spectrum as such. In fact, it is reasonable to suppose that this spectrum can be plotted as a frequency distribution and will be a bell-shaped curve. Where would humans fall on this curve?

Before answering this let’s pay homage to the debate of the Anthropic Principle. This principle states that the universe is geared towards producing us. It is derided by modern scientists because it seems like cosmic hubris. Since the time of Copernicus, we have been moving away from this thinking starting with the heliocentric model of the solar system. In keeping with this trend the latest proposal is the multiverse which solves the problem of fine-tuning of physical constants. This change in thinking is called the Copernican Principle, or Principle of Mediocrity, and would suggest that we are most likely an average way to make intelligent life. We are not found at the tail ends of the bell curve, rather smack in the middle. Earth-like biology is probably a rather easy way to make intelligent life in this universe/multiverse. Applying the Principle of Mediocrity, the frequency of earth-like complex life, is a surrogate marker for the frequency of intelligent life in the universe. That is very important because we can actually say something about the possibility of earth-like life out there. What does science say? How difficult is it to make earth-like life?

If you think we are in an infinite multiverse where all possibilities become actualities, then this question might be of less importance to you. Because even if it’s one in a zillion zillion, there ought to be an infinite number of earths out there in the multiverse. This is theoretical physicist Brian Green’s take on the matter. There is another Naïve Thinker out there but who is actually the President of Mars, but this doppelganger must be almost infinitely far away. If you are going to be this generous with reality, you will run into a problem. If absolutely everything possible is actualized, then God must exist. And, an all-powerful being would also be God of the whole multiverse. Also, the Flying Spagetti Monster would exist, but God would eat it for lunch. Alright, alright come back down to reality now! This escapade proves the point that we should not be too generous. Such bizarre notions of the possible do not respect the elegant universe we can actually observe, and it’s not a multiverse. . . yet. And, if we eventually find we are in a multiverse, it will not necessarily be infinite. How could we even prove that it is infinite?

Barring an infinite multiverse which I think is reasonable, how difficult is it to make earth-like complex life? We will address this in more detail in part 2.