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Do we need expertise to rate the chance of very low probability events? We make implicit judgments about probabilty quite often (is the bus late), but I catch myself struggling to do so with very low probabilities (has the bus broken down). On the one hand, I can tell you the probability of winning the lottery with one ticket is around one in 45million, and could work it out with a calculator if necessary. On the other hand, I feel I may need expertise I lack to judge whether world war three will break out before noon today, though I might say with some confidence that it is reasonably likely to in the next decade.

Does the evidence for low probability events usually require expertise to interpret, in a way higher probability events do not?

The reason I am asking is that I would put the chance of something, a particular thing, let's suppose me being the future king of england, occurring in my life at about one in ten thousand, and I think that I don't know enough to make judgments that fine, such that it is in effect an unknowable chance.

The same sort of thing for very high probability events: the same grain would be involved. I might conventionally phrase that as not knowing the probability it won't happen, and it's just safe to assume it will. Is there any talk about this in the literature on probability?

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    Your question is ill-posed: the semantics of "expertise" played a nasty trick on you! :) You're confident that the probability of randomly and uniformly drawing a single winning ticket out of N is exactly 1/​​_N_ only because you possess the necessary expertise. We need expertise to compute the probability exactly or estimate it optimally, or conclude that it's undecidable, of any event whatsoever, not only those at the extremes of likelihood.
    – kkm mistrusts SE
    Commented Mar 25 at 23:58
  • ok so could the question be reasonably phrased in terms of needing to have some particular knowledge, if we are to assign finely grained probabilities to events @kkm-stillwaryofSEpromises ?
    – andrós
    Commented Mar 26 at 6:15
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    Yes, it indeed is. You either know how to restate the problem in the language of the probability theory and then solve it, and are an “expert”, or you don't, and aren't one. Probabilities don't need to be "fine grained"; I don't even know what it means. Probabilities are exact numbers. You solve for a function p(x) for all x of interest in a subset of the probability space. It may be discrete (a throw of a die) or continuous (the time until a radionuclide decays), but it's exactly defined in either case.
    – kkm mistrusts SE
    Commented Mar 27 at 23:59
  • by finely grained i mean very high or very low, e.g. @kkm-stillwaryofSEpromises or, perhaps, to a certain level of accuracy
    – andrós
    Commented Mar 28 at 0:11
  • This is irrelevant in the end. Mathematically, p(x) is a precise real number in [0, 1] which sums up to 1 for all x in the discrete case or integrates over the set of all possible x (possibly infinite) to 1. I think I understand where you're coming from. While we assert that p(x) exists, in practice there may be no possible way to derive it exactly, so instead we are either fitting experimental data to a p(x) of a known form, or, when we have no idea, use universal estimators (from histograms up to DNNs with billions of parameters). You're thinking about the fringes of these models... (1/2)
    – kkm mistrusts SE
    Commented Mar 28 at 3:04

3 Answers 3

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given the assumption of uniform distribution, the probability of winning some lottery by buying N different tickets is a well-defined arithmetic question/problem/statement of a kind that guesses about buses and wars is not, and cannot possibly be

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  • maybe I should add "unless someone is a maxwell's demon sort of creature, of course", but anyway :p
    – ac15
    Commented Mar 25 at 14:15
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Do we need expertise to rate the chance of very low and high probability events?

Knowing the total number of unique states is a good start. For a simple coin flip there are 2 unique states. A deck of playing cards has 52 unique states, etc. The minimum probability (greater than zero) decreases proportionally with the number of unique states: P_min = 1/N where N is the number of unique states. P_max = 1 - P_min.

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  • that is a little helpful.
    – andrós
    Commented Mar 26 at 7:00
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Your question is fundamentally flawed, leading to a contradiction. Suppose it were true that you needed specialist expertise only to rate the chance of very low probability events. You would therefore need exactly the same expertise to rate the chance of such an event not happening, which would be a high probability event.

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  • interesting, and i think i am compelled to agree
    – andrós
    Commented Mar 25 at 22:01
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    I can tell you the probability of a meteorite not falling on you (or me, or whoever) tonight to ten or more significant digits just by guessing. To tell you the probability of a meteorite falling on you tonight to a single significant digit, a lot of data and modelling (aka expertise) would be needed.
    – Pere
    Commented Mar 25 at 22:36
  • The question is about "very low and high probability events". There's no contradiction.
    – kkm mistrusts SE
    Commented Mar 26 at 0:06
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    @kkm-stillwaryofSEpromises yes, but you are not reading the original version of the question, which was the one I answered.
    – Marco Ocram
    Commented Mar 26 at 6:18
  • Oh, I see. I agree that the question is ill-posed w.r.t. the meaning of “expertise”. Implied is that one needs “expertise” (whatever it is) is required only to compute probabilities extremally close to 0 or 1, but not the rest. You may formally counter that with the case of an N sided fair die with uniquely labelled faces. The prob. p(X) of any outcome X in the event space of N uniform outcomes is 1/N, and may be made arbitrarily small by choosing a large N. My resolution is that expertise is required in any case, for any N: knowing that p(X)=1/N in this problem is expertise.
    – kkm mistrusts SE
    Commented Mar 27 at 23:47

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