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.

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