Here’s your next reading assignment. Read Sections 2.3.7 and 2.4 in your textbook and answer the following questions by 8 a.m., Friday, February 3rd. Be sure to login to the blog before leaving your answers in the comment section below.
- Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
- What’s one question you have about the reading?
1) We are assume that life only occurs at specific temperatures and that the there must be an Earth-like planet orbiting a Sun-like star for there to be some life there. I think that life on on other planets could happen at different temperatures, or with non-Earthy planetary scales, pretty easily. Thus, the astronomically low odds presented here go up considerably.
2) Is there a way to sample from a small population without replacement and estimate your error?
1) Suppose A is the event of life on a planet and B is the event that a planet has just the right temperature orbiting a Sun-like star at the right distance. Then P(B) is very low and one can argue the existence of a planet like Earth with the right temperature, orbiting the Sun at the right distance is very special. Furthermore, looking from a universal perspective, P(A) is also very low. However, P(A|B) or the probability of life on a planet, given that the planet is Earth, is not as low since the Earth provides the right conditions (temperature, oxygen, water, etc) for life to flourish and so the presence of life on Earth is not special (i.e. the presence of Earth itself is special, not the fact that it is teeming with life).
2) A more common expression for Bayes’ Theorem is:
P(A|B)=P(B|A)*P(A)/P(B)
Is this equivalent to one in the book?
1. I agree that the relationship between the our sun’s size and planet distance is special but there are a great number of other combination of star sizes/planet distance that would allow for life on that planet. Also, the universe is so vast, it wouldn’t be surprising if there was another plant like earth with the same sun size and planet distance.
2. Can you do an example of Baye’s Theorem in class?
1) Let even A represent life on a planet and even B represent the right temperature on a planet orbiting a Sun-like star at the right distance. Both P(A) and P(B) separately are low probabilities. But P(A|B) is not low anymore. So this tells us that the presence of life on Earth is not special.
2) I don’t understand why the book states that we need to use Bayes’ Theorem when tree diagrams are difficult. To me it actually makes more sense to use it when I have already constructed a tree; in this case I see things more clearly.
1. Well given that P(earth-like temperatures) is so tiny – which it might not truly be, since we did just find an earth-like planet a few months ago and have all of space and time to sort through – it seems that the chances of life on Earth are small; however, we don’t really know how likely it is that life exists given earth-like temperatures. For any given planet with temperatures similar to that of Earth, the probability of life developing there might be like 99%, so even if P(earth-like temperatures) is low, the P(life developing | earth-like temperatures) might not be as tiny.
2. Nothing this time around really.
1. There are two events considered: existence of life as we know it (A) and the existence of a planet situated for an Earth-like temperature (B). The statements asserts that since A requires B, an extremely small P(B) requires an extremely small P(A). Which if, as is implied, P(A | B^c) = 0, then the statement is true. However, the statement makes nothing about P(A | B).
2. Bayes’ theorem and its use was the hardest part of the reading.
1. The claim lies in the idea that there are three events: (A), a star being approximately the same size as the Sun, (B) a planet being approximately the same size as Earth, and (C) the planet orbiting at a distance approximately that of our distance to the Sun. Theoretically, the probability mentioned is (C|B|A). However, the statement “Life as we know it on Earth” serves as a qualifier – Life could exist in a variety of other environments, but the probability being given is simply that of a near-exact copy of an Earth-like environment. Also, given the “astronomical” number of stars and planets in the observable universe, the “astronomical” odds of finding such an environment are, realistically, almost certain to be fulfilled.
2. Are there limitations on the number of events included in Bayes’ Theorem?
1. Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
Basically, what you are trying to compute is the P(life & planetSize & sunType & rightDistance).
It is fairly well agreed the chances of such an event are quite low. If, however, you consider the number of planets there might be in the entire universe, it may be reasonable to assume there are at least a few planets that meet such criteria.
At that point, it would be more relevant from a “just look how close we’ve come to not existing, now eat your vegetables” perspective to consider the conditional probability of life existing at all given the conditions were right.
At that point you would really be computing P(life | planetSize & sunType & rightDistance), which may luckily be fairly high. However, that would have to be purely an observational study, and we couldn’t well prove anything, now could we?
2. What’s one question you have about the reading?
What situations would Bayes’ theorem be easier than a tree diagram? It seems much more straightforward to make a tree diagram, or at least pretend to make one.
1. I would say that the presence of life on Earth is a lot more than “somehow” special since the probabilities of Earth orbiting at the right distance from the sun to have livable temperatures is so small. Also, there are more factors to life being on Earth other than Earth having the correct temperatures, so the person should take other factors into account, which would make the probability of life existing on Earth even smaller.
2. Do we need to memorize Bayes’ Theorem?
1.
The odds that an earth like planet orbiting a sun like star is astronomical, however, given this the odds of some life developing over 4.3 billion years is not very special given these right conditions. On a side note, given that there are a relatively infinite number of sun like stars and earth like planets in the universe, life should not be unique to earth.
2.
I would like to see more examples of Bayes’ Theorem with more variables.
1. Since life could only exist on a planet such as Earth, the fact that life exists here is not that special. The fact that there is a planet such as Earth, however, I feel is special.
2. Of what use is Baye’s Theorem?
1. The temperature of a planet is a result of innumerable variables coming together. Applying the following events to Bayes’ Theorem, one can see how the probability of another Earth-like planet is an infinitely small number divided by an infinitely large number (where the limit approaches zero).
Assume B is the event that an Earth-like planet exists
Assume A_1 is the event that the distance from the sun is just right
Assume A_2 is the event that the sun is a “sun-like” star as we know it
Assume A_3 is the event that the temperature is just right
Assume A_4 is the event that an Earth-like atmosphere exists
…
Assume A_one_billion is the event that atom # 10^999999 decided to bond with atom # 10^999999 – 1
…
Basically, it can’t happen again. We are pretty special.
2. Can we use Venn diagrams to show conditional probability? Would it just be a bunch of circles within parts of other circles within a sample space?
1) A simplified Bayes’ Theorem says P(A|B) = P(B|A)*P(A)/P(B).
Let A = Position of Earth is such that temperature is “right” for life.
B = Life occurs
P(A|B) = P(B|A) = 1, so P(A) = P(B). If we allow P(A) to blow up, P(B) will blow up too.
It seems like there is another way to think about that seems to refute that conclusion, but I can’t quite put my fingers on it.
I appreciated the thought-provoking question.
2) What kind of applications to Bayesian statistics have?
1. This statement is false. In the grand scheme of things, it is indeed rare for a planet to be able to support life (the odds are “astronomical”). However, in statistical terms, if we define events A and B as:
Event A: a planet is the right distance away from a star and thus has the right temperature in order to support life.
Event B: A planet has life on it.
The probability that a planet has life on it (Event B) GIVEN that a planet has the right temperature to support life on it (Event A) is not very low. Thus, life on Earth is not very special after all since Earth falls into the group of planets which support life and the probability to actually have life when a planet can support it is not “astronomically” low.
2. Bayes’ Theorem uses a really bulky formula. Is there an easy way to remember it?
1) For life to occur, the important component is the temperature. Therefore, a planet does not have to orbit a Sun-like star to achieve the right temperature, just any star with the right distance and radiation. Even though life on other planet seems unlikely, given the population size, the entire universe, that continuously expand, life can exist on other planets since Earth already exist, in the Milky Way galaxy. If assumption is made that Earth is the only planet with life in this galaxy, there at least one Earth-like planet with life on every other galaxy. Distance and size of the universe make it hard to find or communicate with other planet with life on it.
2) Does using Bayes’ Theorem even when tree diagram is possible eliminate the chance for error?
1) This question assumes that the only variable affecting Earth’s temperature is the distance from the sun. There could be other lurking variables such as the strength of the sun’s UV rays or the size of the planet and, by extension, it’s gravitational pull. Having “astronomical” odds is also somewhat misleading. Astronomical, to me, implies a high number, when in fact the probability would be extremely low assuming all the given facts are true.
2) I don’t quite understand why, in 2.4, they first calculated in the raffle experiment the likelyhood of you not winning. Is it not possible to directly calculate the probability of your winning, given that the pool of tickets is decreasing by 1 each time? I tried calculating this multiple ways but I couldn’t come out to the correct answer… Help!
1) Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
> We have only one sample, and that is life that _did_ evolve on earth. We do not know the general conditions for life to evolve on any planet. We can make assumptions, but they will always be just that; assumptions.
2) What’s one question you have about the reading?
> Derivation of Bayes’ Rule
1) I think they’re implying that there was equal probability for the Earth to end up anywhere else in the solar system and thus be the wrong temperature, which is pretty much as far from truth as you can get. Just because there is something like 93 million miles between the sun and the edge of the solar system doesn’t mean the Earth had a 1/93,000,000 chance of forming where it did. It sort of reminds me of the guy who said there is a 50% chance of the planet being destroyed by the LHC because there are only two options.
2) Just to double-check, P(A|B) is NOT equal to P(B|A) right? Unless P(B) = P(A)?
1) I see a tree diagram in the statement.
First branch: probability that the sun developed with “sun-like qualities”
Second branch: probability that the earth is the proper distance from the sun for the correct temperatures, given the sun’s condition
Third branch: probability that the earth is “earth-like,” given the conditions of the sun and its distance from the sun
Fourth branch: probability that life developed on earth, given the criteria in the prior branches.
2) A few more examples would help
1) There are an estimated 100-200 billion galaxies in the universe, each containing billions of planets and hundreds of billions of stars (the Milky Way has somewhere near 50 billion planets and 200 to 400 billion start). Water was recently discovered on Mars, and the Kepler telescope recently discovered over 50 different planets similar in size to Earth, and within a habitable distance from their respective stars. While life as we no it could only exist within a specific set of temperature and environmental regulations, it would be ignorant for us to say that life could only exist on Earth-like planets. But in a universe where life could only exist on Earth-like planets, there are several planets within observable distance to Earth, with similar Earth-like characteristics. This is only within observable distances, if this number were to be extrapolated over the entire universe, there could be billions of Earth-like planets in the universe. This makes the existence of extraterrestrial life almost inevitable. Based on this, I would say that life on Earth is lucky, not special, but coincidental. The odds for life on any given planet may be astronomical, but the amount of planets in the astronomical universe is astronomically astronomical.
The odds of the Earth’s temperature had to be just right , and the odds of an Earth-life planet orbiting a Sun-like star at just the right distance are relatively small. To make the presence of life on Earth happens , we have to multiply the two small odds which is nearly infinitely small.
1. For a planet to have life, if must have the required temperature that is given from our distance from the sun. That means that the P(life given the correct temperature) is very small and P(correct temperature) is even smaller. If you multiply those together you get P(life) which would be incredibly small and very improbable.
2. I just don’t really understand Bayes’ theorem
“For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
Since we’re applying conditional probability, this means we have an outcome of interest and a condition. So what’s our outcome of interest and condition? Well, the outcome is not life, in this case. The outcome is “temperature = habitable”, and the condition is “planet around star”. It’s not P(life|temperature = habitable), it’s P(temperature = habitable|planet around star). These are two entirely different statements. And yes, the odds are astronomical, in that it relates to astronomy. But the odds that there is another large mass of rock orbiting a burning mass of hydrogen gas at a distance that creates temperatures between 273 K and 303 K (typically) is fairly likely. Given the absolute sheer size of the universe, and the fact that we as humans have found many “Earth-like” planets outside our solar system, there is sure to be other hospitable planets.
What is Bayesian statistics, and why is it (apparently) so important?
1. Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
The probability of the temperature is probably very small, with a large chance of having life on a plant with that temperature.
2. What’s one question you have about the reading?
It was pretty clear
1) In general, the more conditions one adds to a problem the more specific the outcome received. In order for us to live on earth we needed the perfect temperature, a star at our center, and to be the perfect distance away from it, among other things. These three conditions provide the outcome, humanity. Without just one of them we would not survive. The probability of all events simultaneously occurring is not high so that makes it more unlikely but still a definitely possibility (clearly since we are here). I feel that does make life on earth special.
2) What number of scenarios will potentially make tree diagrams too complicated and force one to use Bayes’s theorem?
1. The Earth’s probability of reaching the right temperature on Earth is one based on the distance from a sun-like star given that that there is a sun-like star. In order to calculate this probability we would need to find out how many sun-like stars there are and the distances that potentially habitable planets are away from them. Then the calculation would be the probability of a sun-like star and a proper distance divided by the probability of a sun-like star.
I do not agree with the statement, because the earth varies in distance from the sun throughout the year. Thus temperature is not the only deciding factor creating life. Also there have been a few discoveries of planets orbiting non-sun-like stars with a corrected distance that may contain life.
2. Does Bayes’ Theorem work with all tree diagrams? Is there an example where Bayes’ Theorem falls short?
Yes, given the conditions for life are “astronomical” the word special can be used almost synchronously with astronomical. The combination of these conditions are indeed very speical.
And no, the reading was very straightforward
1. Using the Bayes’ Theorem, if we set A1 to be the right distance from the sun and B to be the existence of earth, then plugging these values in the top and bottom cancel and we find that given the existence of earth, the probability of earth being the right distance from the sun is 1. Therefore, the argument is circular?
2. Am I using Bayes’ Theorem correctly?
1)The fact that life does exist is not only based on temperature, it is because we have ozone layer that prevent us from getting UV light from sunlight. However, there is possibility for other life to be existed in other planet because we can’t assume life being as in human, there are always other life being out there like for example on earth we have animal and trees that share the same nature as we did. what I am saying is that different life need different need in order to survive
2)I still don’t understand baye’s theorem. It is confusing
This suggests that life existed first and that the conditions of Earth had to perfectly match the requirements of that pre-existing life for it to be present. However, Earth was here first and life came to exist based on the conditions of Earth. Therefore those “astronomical odds,” in the context that they were used in the quote, are baseless as the probabilities for there being an Earth-like planet for humans to exist on and the probability that the life Earth would produce would be capable of living under its conditions are entirely different.
After reading I still don’t really understand why P(B|A1) * P(A1) is in the numerator of Bayes’ theorem. Probability of A1 given B and that of just A1 don’t seem to cover the possibilities well. Why not P(A1|B) * P(A1|~B) ?
1. The likelihood of life on an earth (presuming the described conditions) is high, but because the conditions are so unlikely, the conditional probably of our life is “special.”
2. How would a three or even more condition tree diagram look and function?
1. The probability of life flourishing on Earth is a statistical improbability because it is dependent on the conditions mentioned in the statement. Since these conditions already have a low probability of happening when unrelated, the conditional probability of life forming on Earth becomes highly unlikely. Conditional probability involves multiplying the probabilities together, and multiplying such small chances together gives a minuscule chance of life-forms actually forming on Earth.
2. Can we really not come up with a better test for Lupus?
1) While the odds of an earth with the perfect temperature are astronomically low, there are an astronomical number of planets in the universe. The odds that only one of these planets has the right conditions for human life is not as low.
2) I understood everything in 2.3.7 up until theorem 2.63. I reread the section a few times and I’m still not sure.
1. The existence of earth is very special knowing that there is a very high probability that there is a planet at the right distance from the sun to create the conditions of earth but there aren’t any.
2. I want to know more about bayes’ theorem
1) Conditional probability doesn’t apply in this case. There are too many other factors and variables that we may not even be aware of. We are assuming that life can only exist on a planet who’s conditions are like earth and also that life can only exist as we know it. I believe these are poor assumptions that will ultimately undermine any statistical analysis at this scale. NASA actually found life last year that differed from our previous knowledge of life as we know it. “But now researchers have coaxed a microbe to build itself with arsenic in the place of phosphorus, an unprecedented substitution of one of the six essential ingredients of life… [on earth] we have a single sample of life. You can’t look for what you don’t know.” (The link is posted below) Who knows what other assumptions we make are also not true? Given this, I believe the odds increase greatly as our understand of life changes.
<http://www.wired.com/wiredscience/2010/12/nasa-finds-arsenic-life-form/
2) It says if we sample a small pop without replacement, we no longer have independence between the observations. Is the opposite true? If we replace does it make the observations dependent on each other?
1. It is not valid to say the presence of life on Earth is special due to the sun being at the right distance from Earth to create Earth like temperatures. This is because although there is a relationship between the distance of Earth from the sun, these variables are all disjoint outcomes.
2. What kind of data/trend can you really make a conclusion about when using small populations for sampling?
1. The probability of finding another Earth-life planet is very small.
2. What is the maximum sample size to be considered as small?
1) Even though the chances are very very low of happening, there is still a slight chance and it just so happened to end up that way. The chances were probably only one branch of a very large tree diagram but it ended up that way.
2) No questions after the reading.
Conditional Probability is thought of in terms of Given A what is the probability of B.
So… Given That we’re thinking about it, the probability of life existing on Earth is 100%
It for A to be true, so must B
1. The statement makes it sound like multiple factors that each have a small probability must all occur together in order to produce life on earth, however, the statement really only addresses the probability of one issue: temperature. It doesn’t seem so unlikely that a planet could reach a particular temperature…
2. In Bayes’ Theorem we often multiply P(A|B)*P(B), but this is equivalen to P(A&B)/P(B)*P(B) = P(A&B). Wouldn’t it be easier to conceptualize if we put all P(A|B)*P(B) terms in the form P(A&B)?
1. The odds may be astronomical to have that combination of planet/star/distance, but once that combination exists I think the odds of life are pretty high.
2. This isn’t about the reading but something in lecture that still doesn’t quite make sense. This week we had the example about the students who made A’s in astronomy and biology. I don’t remember the exact numbers but it was like 80 got A’s in astronomy, 73 got A’s in biology, and 40 got A’s in both. What I can’t seem to wrap my head around is why a sample (student) could be counted in two categories.
It seems to me like the statistics were taken incorrectly (or at least illogically), like you can only belong in one category. Either you got and A in astronomy, OR you got an A in biology, OR you got an A in both. To me it doesn’t make sense why you would count someone in two categories.
although the probability of earth like conditions (small star with rocky planet at the proper distance) are rare, even if they only occur around one in a billion stars, there are hundreds of billions of stars, so earth probably isn’t entirely unique. This assumes only earth like planets can have life. We know of species on earth which are capable of living in extreme environments, so the probability of life elsewhere increases when we look at the possibility of extremophiles on other planets.
no questions from reading this time.
Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
This is stating P(Earth-life planet orbiting a Sun-like star | just the right distance to produce Earth-like temperatures) = a very small number.
It also states the implication that Earth’s temp just right -> life to occur as we know it.
So, we are interested when Earth’s temp is just right. Thus, we want the opposite of the statement above and should use Baye’s formula. This will multiply that probability by every probability B(a1), B(a2)… turning the astronomically small number into a more reasonable number. Thus, the statement (which likely isn’t meant to be entirely mathematical), is not one I’d fully agree with given the aforementioned reasons.
What’s one question you have about the reading?
Reading was fine. One more example using trees could be good, but I thought it was clear.
1. Our probability for survival depends on factors such as food, water and temperature. Earth, being at just the right distance while orbiting the Sun and having just the right temperature makes this statement a conditional probability statement.
2. Can we do more examples of sampling from a small population in class?
1) If the conditions for the earth to have life are so many and have to align so perfectly then the odds are very rare. It would be one great tree diagram that only one outcome at the very end would result in life on earth.
2) Example 2.65 does not make sense to me.
1) The presence of life on Earth is actually not particularly special at all- the statement makes clear that Earth is, by nature, a hospitable planet, thus the presence of life is not unique whatsoever. That a planet in general in our solar system is hospitable is rare, yes, but that EARTH in particular is, is not.
2) No question this reading.
Given what you know about conditional probability, what do you make of this statement? “For life to occur on Earth as we know it, the Earth’s temperature had to be just right. Since the odds of an Earth-life planet orbiting a Sun-like star at just the right distance to produce Earth-like temperatures are astronomical, the presence of life on Earth is somehow special.”
What’s one question you have about the reading?
1) First of all, I like the “astronomical” pun. Secondly I would just point out that while the odds are small, the universe is very large, and has a very large number of solar systems – without a more extensive look at the actual numbers involved, making such a statement is questionable. Finally, I’d point out Bayes Theorem and note that the speaker’s looking at tiny odds in one probability direction but talking about something being special in the other direction (I think? If I’m reading the question correctly?).
2) Is there a situation where a tree diagram is not only difficult, but impossible? Where one would be absolutely FORCED to use Bayes Theorem?
1. Bayes’ Theorem shows us how unlikely this event is. First, call the probability that the Earth is at just the right distance to produce Earth-like temperatures ( which is “astronomical”) probability A, and the chance that Earth-like life came to exist instead of other life (assumed to be extremely small as well since there are many possibilities) probability B1. So,
P(B1|A) = (P(A|B1)*P(B1))/(P(A|B1)*P(B1)+P(A|B2)*P(B2)+……+P(A|Bn)*P(Bn)).
Since n could be extremely large, the denominator would likely be extremely large as well, and given the already low probability in the numerator, the total probability of this event would be extremely small.
2. Are we going to see many more of these more or less proofs (like for Bayes’ Theorem) in this class? It seems more likely in probability than in statistics, and probability seems to be a much smaller part of the class.
1. It makes no sense because its stating that the chance of earth revolving around a sun like star is high, given that the temperature should be right therefore earths place is not special.
2. Can conditional probability apply to multiple givens instead of just 1?
1.) The subject of this argument is the conditional probability P(life on earth | orbital conditions were just right). The argument basically says that because the P(orbital conditions were just right) is so astronomically low, that the presence of life on earth is somehow remarkable. But that ignores the other part of the calculation, the P(life on earth | orbital conditions just right). Although the probability of getting just the right conditions may be very low, the probability of life arising once the conditions are present might be very high (many chemical experiments and physical chemical theories in fact predict that it is so high as to be basically deterministic). In other words, the probability P(life on earth | orbital conditions were just right) may be very high, although P(orbital conditions were just right) itself is low.
2.) How many scenarios would you say is “too many” to draw a tree diagram?
1. Given the relatively infinite size of the entire universe, it is not unfathomable for there to be a similar planet in the universe such that life could exist. In 2009, NASA estimated that there are 10^24 stars in the universe we live in. With each of those stars being the center of their own solar system, the possibility of rock coming together and forming a planet the correct distance away from that star for the temperature to support life is probably not as small as we would think. As far as theoretical physics go, there can be as many as 11 stable multiverses in all of existence, each with an equal number of stars to our universe, which would then mean approximately 11 times the number of stars total that are in our universe. In short, no, from what I know about conditional probability, I don’t think the presence of life on Earth is “special” in the sense that this is the only place where life can exist.
2. Everything seemed relatively straightforward. I am a bit iffy on the relation of the direct relation of Bayes’ Theorem and the tree diagrams, but that might be because they’re not related that much anyway.
1. the probability of having a planet like earth is astronomically low.
2. Baye’s Theorem is still confusing.
A)
From my view point, It is true that the earth is a special for the human (specially). Not just because the sun distance and the temperature. They are many other factors (which some of them are known and others are not known to us). Because of almost infinite factors, which makes earth are suitable to us, the earth is special.
However, from math view point, it is not that special because of the having a place like earth (where the star (ex, sun) is at right distance from the a planet) is can be (the universe is very huge).
B) nothing much
1. I believe that is more special that is much more phenomenal that the earth even is in existence rather than there being life on earth. I think life may exist on other planets but none are actually built like earth.
2. I am slightly lost on how to use Bayes Theorem
1. This probability does make sense, that for the Earth to be condition1 is P1, and condition2 is P2, then the probability of Earth being like this is M = P1*P2*…*Pn, where n is the number of conditions. So, for this large number M, we would expect if M about of Big Bangs happened (ignoring other cosmological phenomenons), then there would be only this 1 that the Earth got right out of all of them.
On an unrelated and more of a meta-physical level, I think this statement is absurd. Every observation requires an observer, obviously. Modern physics even tells us that with thinks like the speed of light and time dilation. But let’s say that our solar/galaxy conditions were not this way and the Earth did not harbor life…who would observe it? There would be no observation, there would be know ‘we’ or ‘know’. For us to ‘know’ these conditions, there is no other possible outcome.
2. In chapter 2.4 it mentions that this only applies to small population sizes. What about large populations? I assume that replacement isn’t an issue.