Surprising results in probability

Following an interesting discussion on my post on UFO cults about the likelihood of life having emerged somewhere in the universe, I thought that I would explore some non-intuitive results in that case and others involving probability.

I have mentioned before that it is hard to get an intuitive idea about probabilities and that it is easy to get led astray. The Gambler’s Fallacy is one example. In that case, when the outcome depends on the roll of the dice or where a ball on a roulette wheel lands, people tend to think that, in the case of the latter, several blacks turning up in a row makes the next outcome more likely to be red. This is not true because each event is independent of the others. It does not matter how many blacks turn up in a row, that does not change the probability of the next one. Many a gambler has been ruined by thinking otherwise.

A converse situation in which the events are not independent can lead to another incorrect expectation. A famous example is to ask people what is the probability that in a room containing 23 people, two or more of them will have the same day and month of birth. Since the number of people is small compared to the number of days in a year, it is tempting for people to to think that the probability would be small too. But that is true only if you pick one person and then calculate the probability that someone else has that very same day.

If you do not specify the day in advance, then the calculation is different. You do it by picking any one person and asking what is the probability that the next person will not have the same birthday. That probability is 364/365. Then you take a third person and ask what is the probability that that person’s birthday will fall not fall on either of the other two days. The probability that it will not fall on the first person’s day is 364/365 and that it also will not fall on the second person’s day is 363/365. Hence the probability that it will not fall on either of those two days is (364/365)x(363/365). If you continue on in this way for 23 people, the probability that the last person will not share a birthday with any of the earlier 22 people is given by (364/365)x(363/365)x…..(344/365)x(343/365), which turns out to be 0.49 or 49%. Hence the probability the 23rd person will share a birthday with one of the other 22 people is 51%.

In my post on UFO cults, I said that there are estimated 1022 stars in the visible universe and since we know that the probability of any form of life emerging in a solar system is not zero (given that life has emerged on Earth), then it is likely that life has emerged elsewhere as well. I did not however provide a number for a value of p, the probability of life emerging in any given solar system, since we do not know enough to do so. The various attempts to get a numerical value for p (such as using the Drake equation) involve highly speculative estimates that can very over several orders of magnitudes. But we can be assured that it is very small.

Commenter Silentbob cast doubt on my conclusion that it is likely, despite the smallness of p, that life has emerged elsewhere, saying that the probability of life emerging in any given solar system is so small that even having such a large number of stars does not change the calculation much. He suggested that p might be as small as 10-24 and argued that this implies that we would be the only intelligent life even in a universe that is 100 times the visible universe, saying “If those are the odds, then all other things being equal we are not only the only intelligent life in the visible universe, but in a region of space one hundred times the volume.”

I thought that it might be illustrative to look into this claim more closely by doing a calculation.

We start by noting that if the probability of life emerging in any given solar system is p, then the probability that life will not emerge is (1-p). If there are N (=1022) stars, then the probability that life would not have emerged on any of them is given by P=(1-p)N. To calculate this quantity for very large N and very small p, we use the fact that e-z can be written as the limiting value as N→∞ of the expression (1-z/N)N. If we take p=10-24=1/100N, then P=(1-p)N=(1-1/100N)N=e-1/100=0.99. i.e., the probability that life has emerged on one of the 1022 stars is about 1%. This is still small but not insignificant.

Since p is unknown and could be off by several orders of magnitude, what if we take it to be given by p=1/N or 10-22? This is still a tiny probability but now P=0.37, meaning that there is a 63% probability that life emerged elsewhere. The same probability of life occurring being 63% is obtained if we take Silentbob’s example of p=10-24 and N=1024. So far from being so highly unlikely as to be effectively zero, even with those numbers, the emergence of life becomes very likely.

All these are speculative because we do not have a good way to estimate p. But the point I wish to make is that things which on the surface seem very unlikely (like a random group of 23 people sharing a common birthday) can, when actual probability calculations are carried out, turn out to be likelier than one thinks.


  1. Matt G says

    When I got my first Apple device (iPad) back in 2012, I decided to put a quote in the signature for outgoing mail. I chose a version of: “It is true that may hold in these things, which is the general root of superstition; namely, that men observe when things hit, and not when they miss; and commit to memory the one, and forget and pass over the other” by Francis Bacon. When you unexpectedly run into someone you know while on vacation, do you think about how many times you did NOT run into someone unexpectedly while on vacation?

  2. Rob Grigjanis says

    Since the number of people is small compared to the number of days in a year, it is tempting for people to to think that the probability would be small too.

    The key insight is that the problem isn’t about individual people. It’s about pairs of people. And for 23 people, there are 253 distinct pairings. 253 is not much smaller than 365.

    Hence the probability the 23rd person will share a birthday with one of the other 22 people is 51%.

    I know what you mean, but that’s misleading. The probability that any particular individual (the 5th, 11th, 23rd, whatever) shares a birthday with any of the other 22 is much smaller than 0.51. You’ve calculated the probability that there is at least one pair which share a birthday. And most pairs don’t include the 23rd person (the 23rd person (or the 5th. etc) is only in 22 of the 253 possible pairs).

  3. robert79 says

    Regarding the Gambler’s fallacy, if in fact you believe the roulette board may be biased, but are unsure whether it’s biased towards black or red, several black outcomes in a row would in fact lead you to conclude that the next roll is (ever so slightly) more likely to be black.

  4. sonofrojblake says

    I thought that it might be illustrative to look into this claim more closely by doing a calculation.

    It was indeed illustrative. It illustrated extremely clearly how wrong Silentbob’s “thinking” was. Thanks for taking the time, it was interesting. I doubt they’ll learn from it, though. /shrug/

  5. robert79 says

    What @2 Rob Grigjanis is correct, this is com[pletely unrelated to the birthday paradox.

    You can either:
    -- ask what the probability is that two people in your class have the same birthday.
    -- say “it’s my birthday today!” and ask if it’s anyone else’s birthday.

    Those are two different probabilities. In the latter case you’ve selected for the situation where it’s already your birthday.

    Similarly for intelligent life, you can either:
    -- ask what the probability is that two planets (galaxies, visible universes, etc…) both have life
    -- say “I’m alive” and ask if anyone else is out there

    In the latter case, that may be a *HUGE* selection effect.

  6. Rob Grigjanis says

    The same value of P being 63% is obtained if we take Silentbob’s example of p=1e-24 and N=1e24.

    To be fair to Silentbob, his example (if I read him right) was p=1e-24 and N=1e22. As you calculated, the probability of any life occurring in the observable universe is then about 0.01. You say that’s not insignificant, but it does mean any occurrence of life in the observable universe is unlikely.

  7. birgerjohansson says

    I did not need to revisit my university statistics course to accept a low probability for complex (or even sapient) life in our galaxy (as distinct from the entire universe).
    As I have mentioned in previous threads, D. Waltham’s ‘Lucky Planet’ provides a long list of hurdles for a planet to overcome to remain hospitable for complex life during a four-billion year time span.

    Yes there will be plenty of other places where life has produced sentience. But not in our corner.

  8. Silentbob says

    Yes, you’re correct I did the “birthday fallacy” of assuming that if there is a 1:365 chance of an individual having a particular birthday, we can expect one person in a group of 365 with that birthday. X-D

    That was dumb.

    But I was not talking about independent probabilities, like a roulette wheel landing on black for a fifth time. But rather the existence of an environment where casinos can exist, and the probability of the existence of a casino, and the probability of the casino having a roulette wheel, and the probability of it landing on black.

  9. file thirteen says

    Probability mathematics is useful when the probabilities are known, or even the probability of the probabilities can be reasonably inferred. Unfortunately when the probabilities themselves are so unknown as to be just speculative, any results of processing them are meaningless. Garbage in, garbage out.

    Best to concentrate on the details until we have good estimates for them, at least. We know one variety of life can exist, Earth’s. So putting aside the speculation of other types, what is the probability of other “Earths” being out there (which you can break down into frequency of similar size, composition, being in the “habitable zone” for liquid water to exist etc., all of which have chances that can be better estimated).

    Then switch back to the issue of how life arose here. Without knowing how likely it is that compounds can “evolve” (it’s a similar process to evolution) to other ones, it’s not really possible to make a realistic estimate of how likely a planet like Earth is to develop life. But I’ve heard amino acids have been spotted in interstellar dust clouds, so at least the building blocks seem to be plentiful enough -- if there are enough “Earths” around. But I don’t think we will be able to make meaningful probabilities of even our type of life arising elsewhere until we work out how ours probably did.

  10. Mano Singham says

    Rob @7,

    If you read the part I quoted from his comment in my post, you will see that he argued that even with a universe 100 times the volume of the visible one (i.e., with 1024 stars), a value of p=10-24 would result in almost zero probability of life emerging.

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