playing with probability

Reason Truth

What is the probability of any one exclusive religion being the correct one (“exclusive” means: claims to be the only true one)? This question popped into my head the other day and made me think that you could theoretically calculate that probability using the following formula:

n-(n-1)/(n+1) or 1/(n+1)

Where “n” is the total number of exclusive religions.

Thus, you subtract all but one of the religions from the total, then divide by n+1 to account for the fact that they could all be wrong.

So, if we put together a list of exclusive religions like the following, we can calculate the probability of any one of those religions being the correct religion:

  1. Catholicism
  2. Orthodox religions (Greek, Russian, etc.)
  3. Mormons (LDS)
  4. FLDS
  5. AUB (Apostolic United Brethren)
  6. Jehovah’s Witnesses
  7. Seventh-day Adventists
  8. Branch Davidians
  9. Aum Shinrikyo
  10. Old Order Amish
  11. Islam (Sunni)
  12. Islam (Shi’a)
  13. and so on (thousands of small Protestant groups)

These were just the first 12 exclusive religions that popped into my head. So, let’s just say, for the sake of the illustration, that there are only 12 (there are probably thousands, but we’ll keep it simple). Thus, we end up with:

12-(12-1)/12+1 = .0769 or a 7.7% chance

Thus, if there were only 12 exclusive religions, if you assume each had the exact same probability of being right (and therefore, of being wrong), that probability would be about .0769 or a 7.7% of being correct or a 92.3% chance of being wrong. Of course, there are many more than just 12 exclusive religions. But this does seem to suggest that any exclusive religion has a very low probability of actually being correct. For instance, if we assume 1,000 exclusive religions, then there is a .099% chance that any one of those religions is correct or a 99.9% chance of being wrong.

Of course, such calculations are problematic because there are complications that arise when you examine closely related religions or religions that splintered from other religions (decreasing the odds of those being correct). They may be problematic for other reasons.

Any thoughts on whether this is a reasonable way to calculate the probability of any one religion being correct?

The obvious implication is a cost/benefit analysis: Is the cost of belonging to a religion like Mormonism worth the risk of being wrong, especially if the risk is something like a 99.9% chance that you are wrong?

15 thoughts on “playing with probability

  1. Well, I don’t think you’d want to assign equal probability to each group – is Scientology really as likely as, say, Deism?

    You might want to weight by the total number of adherents, but there’s a lot of path dependence. People tend to be the same religion as their parents, so religions that have been around longer tend to have more members.

    I think you could make a serious case for using growth rates from conversion (not birth) because it proxies for plausibility – to convince people to switch churches, you must convince them that it’s more likely that yours is true.

    This is not totally accurate, either, because people participate in religion for social reasons as well as dogma. Plus about a million other things I haven’t thought of.

  2. Maybe the first step would be to establish the probability that there actually is a true religion versus the probability that there is no true religion?

    Chris,
    “Well, I dont think youd want to assign equal probability to each group is Scientology really as likely as, say, Deism?”

    As an atheist, I consider the probability of either being true to be vanishingly small, so in practical (if not mathematical) terms, I would say Scientology is just about as probable as Deism.

    “I think you could make a serious case for using growth rates from conversion (not birth) because it proxies for plausibility to convince people to switch churches, you must convince them that its more likely that yours is true.”

    I think that would just be a form of argument ad populum. It would only tell us how plausible people find the religion. It would tell us nothing about how “true” it is.

  3. Kuri,

    Assuming that the probability is approximately zero that any given religion is true begs the question of the post, which was to figure out the probability (or relative probability) of truth. I agree that Scientology is almost certainly false, but I do so based largely on introspection, not numerical analysis.

    I am not, in fact, making an argument from popularity – I’m not asserting that a religion must be true simply because it’s popular. In fact, this is the very reason I rejected using total membership as a measurement in the first place.

    I don’t want to spend a lot of time defending what was, in fact, an extemporaneous proposal. My basic idea is that you could use the aggregated knowledge of informed people to make some kind of measure of relative truth. It’s kind of like the Iowa Electronic Markets. Suppose there are more Mormon converts to Catholicism than Catholic converts to Mormonism. This is evidence that Catholicism is more convincing that Mormonism. It’s not conclusive evidence and it would be a stretch to put a number on it (“2.3 times as likely to be true”), but it’s informative.

  4. Chris,

    I agree that there are lots of other factors that complicate any simple formula like the one I provided above. But is it possible to simply calculate a generic formula?

    I’m kind of inspired by the formula used to calculate the probability of life on other planets (http://en.wikipedia.org/wiki/Drake_equation). That formula isn’t used to calculate the probability of life on Jupiter (very low) or on Earth (very high), but on all planets, generally.

    So, the question is, is there an actual formula that someone could use to calculate the probability of any one religion being correct?

    My formula assigns equal weight to all “truth” claims: the crackpot street preacher who is probably insane gets the same weight as the Pope or Mohammed. Is that too problematic? I don’t know – ergo, the question.

  5. Chris,
    I’m not trying to be argumentative. I know your suggestion was ad hoc and not something you want to spend hours defending. I was just giving my reaction to it. I simply think that how many people find something how convincing has little relevance to anything except how many people find that something convincing.

    Anyway, regarding “the question of the post, which was to figure out the probability (or relative probability) of truth,” my main point was that I think profxm’s post assumes that there is a true religion. I think to begin to get at the “probability of truth,” we’d also need to include non-religious belief systems, whether rational, metaphysical, or other, as well as hypothetical “unknown” belief systems.

  6. Profxm –

    The Drake Equation itself is mathematical fact – it’s just an application of the Law of Large Numbers. The difficulty with that equation is that the parameters (like the unconditional probability that a planet can support life) are completely speculative.

    What you’re trying to do here is less like the Drake Equation itself and more like a discussion of one of its parameters. What’s the probability that life occurs on a planet? Suppose we knew that there is no life elsewhere in this solar system. Would we want to assert that the unconditional probability of life appearing on a planet is 1/8 (just the sample mean, ignoring Pluto)? Probably not. We’d want to do something more sophisticated. What? I don’t know – ask the astrophysicists.

    I am, however, an economist, so I know how my discipline tends to evaluate matters of fact based on collective behavior. That’s the frame of reference I have.

  7. OK – I think I need to cool down my online persona a little bit.

    First, I think that what profxm is trying to do is argue that no one should, ex ante, be terribly confident in their religious stance. I agree. In fact, it was this type of exercise that first made me doubt the existence of God.

    Second, my commentary above is basically methodological while I think Kuri is focused on the question itself. I’m sorry if I’ve been talking past you instead of addressing your points.

    There is no good way to measure probabilities of events that we don’t get to see over and over again (Feynman makes this point very well in his criticism of the Challenger explosion). I’m just offering one measure of probability – a sort of aggregation of individuals’ subjective probability beliefs.

  8. So, Kuri, you’d suggest including all (1) exclusive, (2) non-exclusive, and (3) non-religious philosophical and metaphysical belief systems into the “n” of the equation. Based on that approach, your “n” could easily push up to around 10,000 or more, making the probability of any given religion being the “true” or “correct” one less than 1 in 10,000. I’m not opposed to this, but the reason I opted for exclusive religions rather than non-exclusive (or inclusive) was because inclusive religions allow for the possibility that many other religions may be correct, which is harder to control for mathematically.

    Chris, if we do assume that I’m asking about one of the parameters, how does that change the equation? I would think that, as an economist, you might have the tools to put together a mathematical model that is more complex than my simple equation. Perhaps you could put together something more complex, as you were reasoning it out, but you’d have to be very careful in determining the important parameters.

    So, for instance, you could include age of the religion as a parameter. But, just like size of a religion, that is problematic. As Kuri pointed out (and I know you recognize this), both of these are logical fallacies: argumentum ad populum (argument from popularity) and argumentum ad antiquitatem (appeal to tradition) are both logical fallacies and do not increase the probability of an argument being right. So, the question becomes, what variables could you include that would either increase or decrease the probability of a single religion being the correct one? The only one I can think of would be a “logical consistency” variable, but that would be virtually impossible to construct as it would require a very clear understanding of the cosmology and theology of every single religion with a ranking based on philosophical principles of logical consistency.

    Other possible parameters?

    Based on your comments, the new equation would be:

    belief system being true = (1/n)*x

    Where “n” is the total number of religious, metaphysical, and philosophical positions and “x” is a logical consistency score ranging from 0 to 1 that is derived by enumerating the theological tenets of a religion or elements of a philosophical position and evaluating them based on their logical consistency.

    Based on this revised formula, I’d say that any given position’s likelihood of being true is basically zero.

    Frack! I think we just offered a mathematical proof of postmodernism! 🙁

  9. profxm,

    I guess we’d have to narrow our “n” down to exclusive belief systems, but I think we should include all types, given exclusiveness. Either that, or find a function for the probability of truth for religious versus non-religious belief systems.

    But we could also take some shortcuts. For example, rather than including “The world exists only in my mind,” or your mind, or each other individual’s mind, which would instantly raise our “n” to 7 billion, we could use “The world exists only in someone’s mind” as one belief system.

    I also think this is a very interesting question: “The obvious implication is a cost/benefit analysis: Is the cost of belonging to a religion like Mormonism worth the risk of being wrong, especially if the risk is something like a 99.9% chance that you are wrong?”

    Somewhat along the lines of Pascal’s Wager, we could look at the costs of not belonging to a given religion. I would argue that Mormonism, for example, has a low “not-belonging cost.” If Mormonism is true, and I faithfully practice some other religion, at worst I’ll end up in the Terrestrial Kingdom. Even if I’m a rank “sinner,” at worst I’ll probably end up in the better-than-earthly paradise of the Telestial Kingdom.

    OTOH, if fundamentalist Protestantism is true, no matter how faithfully I practice some other religion, I’ll be tortured in Hell for eternity if I don’t “accept” Christ before I die. So the “not-belonging cost” is much higher than for Mormonism.

  10. It would probably take over an hour to count those, but I’m going to go with somewhere between 500 to 1,000 if you also include all the various denominations of Christianity, which are listed here:
    http://en.wikipedia.org/wiki/List_of_Christian_denominations

    Ergo, even given the most liberal estimates, you have less than a 1 in 500 chance of your religion being right.

    The question then turns to an issue of cost/benefit analysis: If you have a 99.998% chance of being wrong, how much does your religion have to promise you in order to make it worth taking that chance?

    I wonder if there is any information on this in economics and game theory. Chris? Any sense on how willing people are to bet money when they know they have only about a .002% chance of winning. I guess people who play the lottery have lower odds than that, but there is also a minimal cost – a couple bucks. Does (1) religion promise more than the lottery and (2) is the cost to play higher? My guess is: (1) Not really, and (2) yes. Maybe this explains why more and more people are leaving religion?

  11. Well, theoretically, if the theoretical gains are infinite, then any chance whatsoever that you would get the gains makes it worth betting any less-than-infinite sum.

  12. No, I don’t think this is a reasonable way to calculate the probability of “any one” religion being correct. Various approaches to this are possible, but it doesn’t seem reasonable to me to expect to get an answer to this question based on pure mathematics with no empirical input. It reminds me of Laplace’s discussion of the probability that the sun will rise tomorrow, so in a sense it’s in good company, but I don’t think anybody would accept Laplace’s calculation as valid today.

    The calculation is, however, a perfectly correct derivation of the probability that a religion chosen uniformly at random from the list will turn out to be true, given that they are mutually exclusive and (exactly) one of them is true. Given the “none of the above” option that is implicitly on the list, assuming that at least one of them is true seems quite justifiable. Mutual exclusivity would require that the claims of the various religions be sufficiently well defined to make their truth or falsity be decidable in principle, which may be more than is realistic to expect. Also, any two choices would have to be mutually contradictory, which probably means some insufficiently belief-prescriptive religions (e.g., Shinto and Buddhism seem to coexist without conflict) could not be included.

    I agree with what I take to be the underlying real-world point, namely that the sheer number of religions makes it difficult or impossible for any individual to investigate their truth claims thoroughly. The LDS view seems to be that everyone who has ever had contact with it should spend an entire lifetime (or more), if necessary, to get a testimony. Viewed from the perspective of this discussion, that seems like an over-allocation of time and attention to too narrow a spectrum of possible truth.

  13. Jonathan, that makes sense. 1 in 500 chance of infinite power, knowledge, and immortality. The cost – 10% of your income and about 1/15th or so of your life. Hmmm… Still seems like a bad deal to me, but I can see the appeal.

  14. somehow, i’m thinking one would have to do a probability based on claims, rather than based on number of religions. (It doesn’t make sense to pit all the exclusive religions together, say they are equiprobable, and then throw them over the probability that they are all incorrect.)

    Rather, going back to what Chris had said and Kuri had responded…well, I do believe Deism is much much much more probable than scientology? Why? not because deism is less “strange” (this is speculative, although I do feel it is less “strange”)…but rather, because deism makes less claims. Einstein’s pantheism makes a whole lot less claims (and that’s why I sometimes get annoyed when people try to use his words — despite his personal belief that a personal God was dangerous — and try to use his pantheistic beliefs to justify a personal god) than the creatio ex-nihilo, tri-omni triune God of Christianity.

    so, I mean really, you’d take the probability of each claim (however you figure *that* out), and then to find the probability that a particular religion were true, you’d multiple. p(God) x p(Hell) x p(etc.) The multiplication of several decimal numbers would make smaller (unlikelier) numbers.

    but this all is really speculative no matter how you look at it.

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