How does statistics deal with calculating the odds of an event occurring that seems impossible.

Question

So my thoughts on this lead me to believe that this may be a philosophical question as much as a mathematical question. That said, I find it hard to believe that no mathematician has ever contemplated the issue. My hypothetical will sound silly at first but please read through the question because it does raise a serious question.

If one claims that if I can snap my fingers once I will transform into an elephant, and if I clap my hands I will turn into an elephant for a minute then a lion, how would statistics deal with calculating the relative odds of these two occurrences?

One thought is that the odds of the elephant transformation occurring alone is greater than both the elephant and lion transformations. I think at first this seems logical because the odds of two events happening, more so in sequence, are generally considered less likely than one of the events. However, after further reflection, I think this answer is wrong.

My reasoning is that statistics is a science and therefore its assertions must be based on scientific data. There must be some standard of evidence before anything is accepted as possible. I think we will all agree that there is zero evidence in all of human history to support the proposition that either event is possible. Further, all applicable fields of human knowledge would support that it is impossible for either event to happen. Thus, IMO, absent any evidence to contradict all applicable human experience, statistics, as a science, must conclude that both the elephant event alone and the elephant + lion event must be considered to have an equal 0% chance of occurring.

While the above hypothetical is silly, this question has real applications in philosophical debates, which I would rather not specifically detail to eliminate the odds of creating bias.

Thank you for your consideration.

EDIT 1:

Thank you all very much for commenting! Obviously, some very intelligent people have afforded me some of their valuable time.

I read all of the comments more than once and I must admit, it’s clear that all of the commenter’s proficiency with mathematics so far outpaces my own that at times I find myself struggling to understand some of what is written. My proficiency is with debate and reason, which is why I came here for guidance.

Nonetheless, I feel fairly certain I am not getting the answer that must exist, but rather advanced mathematical analysis. My fault I’m sure so let me rephrase my question.

My judgment and thinking on this subject persuades me that there must be a mathematical axiom that directly deals with my issue. I see now that I should have made that point clear. My thinking is as follows:

Everyone can remember the classic example of the odds of any outcome in rolling a true six sided die, one in six. To my thinking for that to be true certain outcomes must be disregarded because they are impossible by all human experience, such as the die turning into a butterfly and flying away, or the rules of gravity failing to force the die to the floor.

If such silly outcomes, and countless others, were not eliminated from consideration, the odds of any particular outcome of rolling a six sided die would not be one in six but something less. So based on this thinking, my question is what is the mathematical axiom that dismisses from a statistical equation these silly outcomes that defy all human experience.

Apologies if I defer from speaking in mathematical ease on this issue but (obviously) I do not have mathematical expertise so I can’t really think well in that language. Nonetheless, my reasoning tells me there must be a math/scientific axiom that addresses this issue.

EDIT 2:

So Lord Shark the Unknown, user296602, Siong Thye Goh, r.e.s., and BruceET have put a "hold" on my inquiry because it is "off topic."

Please explain why so I can respect your opinion and change my inquiry appropriately.

Thanks!

EDIT 3:

Ok so it seems that the answer is that statistics works with the assumptions one brings to the equation and does not extend to evaluating the quality or veracity of those assumptions or the methodologies by which those assumptions are drawn.

Thus, it appears my question is more about scientific method.

Thank you all for your time and consideration.

Answer 1

My reasoning is that statistics is a science and therefore its assertions must be based on scientific data.

That is a false premise from which you will draw false conclusions.

There is nothing in the theory of probability or in mathematical statistics that states that a die will land on any specific side with $1/6$ probability, or that the die will not turn into a butterfly before hitting the ground. What probability and statistics will let you do is to draw inferences from the hypothesis that a particular die is fair, which is defined in terms of the probability of each side being selected by a process called "rolling" or "throwing" the die.

For example, one of the things you can do with statistics Is to try to detect that a die is "weighted" so that its faces do not each appear with equal probability. In order to make sense of that exercise, we must recognize that not all dice are fair.

If you would care to hypothesize that the die turns into a butterfly with a certain probability during the throw, you can draw statistical inferences from that hypothesis.

Statistics is a tool often used in scientific work, because it enables drawing inferences from scientific models and data. Therefore people who are experts in statistics sometimes end up doing science. In that context, however, statistics is a tool that people use in order to further develop science from scientific data. This is an application of statistics, not a set of assertions of statistics. The actual assertions of statistics (its theorems) have absolutely no scientific content. If they did, they would not be theorems.

Answer 2

Answer to revised question

Yes— when we make declarations of probability, we are implicitly acknowledging some possibilities and excluding others. The reason is that all probabilities are implicitly defined relative to certain background information, e.g. axioms or models of the world or knowledge or assumptions.

We often make statements like “I have just flipped a fair coin. The probability that it has landed heads up is 1/2”, but this is really an imprecise shorthand because the background information hasn't been explicitly specified.

To see this, consider how the right probability assignment might differ based on other information you might have: If you had recorded the trajectory of the coin up until the last few moments before it landed, that additional knowledge would make you much more confident about how the coin would land. Or, if your friend had seen how the coin had landed but you didn't see, you would make an estimate of around 1/2 probability, while your friend would probably make a much different one.

The laws of probabilities are rules about how to assign probabilities based on a given fixed set of background knowledge and assumptions. For this reason, probabilities are not physical attributes of objects in the world. They are not properties inherent in dice or coins — the correct probability assignment is determined as an exact function of your state of knowledge, background model, and so on.

So, you're right: when we say that when we roll a six-sided die, the odds of any particular face coming up is 1/6, we are implicitly assuming a particular abstract model of the world that leaves out many unusual possibilities. In this model, we know nothing about the trajectory of the die — and so the argument from symmetry takes over, saying that each face is equally likely. We assume that the ambient temperature, gravity, pressure, landing surface, etc. are normal, so that the die does not collapse, melt, float away, get wedged in a crevice at an angle etc. In short, we assume a digital abstraction in which the only outcomes are that one particular face will come up. Because we are in a state of complete ignorance about the trajectory, arguments from symmetry uniquely determine the probability assignment in this case as 1/6.

If you wanted to include other possibilities, such as a coin landing on its side, you could compute probabilities for that world-model as well.

---

Answer to original question:

From a Bayesian perspective, you have various candidate hypotheses which we might vaguely describe as:

$$H = \text{magic-seeming transformations cannot occur}$$ $$H^\prime = \text{magic-seeming transformations can occur}$$

Presumably, your current belief places a high weight on $H$ and a low weight on $H^\prime$, though perhaps you can imagine a sequence of things you could see, hear, read about, etc. which would convince you that $H^\prime$ is actually true.

In particular, if you think seriously about the possibility, you might come up with more or less specific possibilities for systematic rules governing the phenomenon:

$$\begin{align*} H^\prime_1&=\text{people can transform into animals}\\ H^\prime_2&=\text{some people transform into wolves during the full moon}\\ H^\prime_3&=\text{contact with enchanted DNA can transform you into an animal}\\ H^\prime_4&=\text{some people can become an elephant at will}\\ &\vdots \end{align*}$$

These fanciful suggestions notheless suggest causal frameworks or working hypotheses for thinking about magical possibilities— and they have different implications for whether you are likely to see someone transform into a wolf (more likely under $H^\prime_2$ than $H^\prime_4$) or an elephant (more likely under $H_4^\prime$) or some other animal entirely (less likely under $H_2^\prime$ and $H_4^\prime$ than with a general mechanism such as $H_3^\prime$.)

In this framework, you might consider it less likely that someone could transform into a lion and into an elephant, because this is a conjunction of two events. Or you might consider it to be more likely that someone could transform into a lion and into an elephant than just an elephant, because you consider a hypothesis like $H_3^\prime$ to be more plausible than $H_4^\prime$. Or you might think that the proposition that someone can become an elephant is strictly impossible, in which case there's nothing wrong with saying that the proposition that someone can become an elephant and become a lion is equally impossible.

In other words: scientific reasoning and the search for working theories can be applied to any phenomenon in the world, including unexpected, counterintuitive, bizarre phenomena such as animal transformations or quantum mechanics.

Most of the time, people do not bother thinking about hypotheses like $H^\prime$ simply because its probability is dominated by other more likely hypotheses' such as $H^*_1$ = someone is tricking me into believing they can become an animal, or $H^*_2$ = I am seeing impossible things because I am actually dreaming, or sleep-deprived, or ill.

Answer 3

I find it difficult to answer your question in terms of hypothesis testing. You are considering an event that many people (perhaps including yourself) feel is essentially impossible. A sensible approach is to make some attempt to quantify your doubts, and to do so in a context that allows for revision of your opinion as evidence accumulates against the possibility of the event--or perhaps even for its possibility. I will illustrate how a Bayesian interval-estimation framework would work in this regard.

Let $\lambda$ be the Poisson rate at which such events occur per 'unit time'. Presumably, you believe $\lambda$ to be very nearly 0. (If you insist $\lambda = 0$ exactly, then the issue is settled in your mind, and there is no possibility of a Bayesian framework for your question.)

Prior distribution. In a Bayesian framework, $\lambda$ is taken to be a random variable--not an unknown fixed constant. We begin with a prior distribution that describes your personal opinion about the possibility of this event, in terms of a prior distribution on $\lambda.$

There is no uniquely correct choice for the prior distributin. One possibility might be $\mathsf{Gamma}(\alpha_0=.002, \kappa = .001),$ where $\alpha_0 > 0$ is the shape parameter and $\kappa_0 > 0$ is the rate parameter. This distribution has the kernel (density function without the 'constant' factor) $p(\lambda) \propto \lambda^{\alpha_0 - 1}e^{-\kappa_0\lambda},$ for $\lambda > 0$ and where the symbol $\propto$ (read 'proportional to') indicates that the kernel does not necessarily integrate to $1$ over $\lambda \in (0,\infty).$

This prior distribution puts most of its probability very near to $0$. Specifically, $P(\lambda < 0.001) = 0.9739,$ as computed in R statistical software. If you are even more doubtful about the occurrence of the event, you could choose values of $\alpha$ and $\kappa$ to give an even larger value of $P(\lambda < 0.001).$

pgamma(.001, .002, .001)
## 0.9738676

Data and likelihood function. Now suppose you run a series of $n = 100$ experiments each over the the 'unit of time' specified for $\lambda.$ Let the number of occurrences of the event for the $i$th experiment be $x_i$ and $t = \sum_{i=1}^n x_i$ be the total number of occurrences. The Poisson likelihood function corresponding to these data is $$p(x|\lambda) \propto \prod_{i=1}^n \lambda^{x_i}e^{-\lambda} = \lambda^t e^{-n\lambda}.$$

Posterior distribution. Bayes' Theorem says that the posterior distribution $p(\lambda|x)$ is the product of the prior distribution $p(\lambda)$ and the likelihood function $p(x|\lambda).$ Because the gamma prior and Poisson likelihood are conjugate (mathematically compatible) it is easy to find the posterior distribution. [Without conjugacy, finding the posterior distribution can sometimes be a daunting problem in integration, perhaps requiring numerical methods.]

Here, the posterior distribution is

$$P(\lambda|x) \propto \lambda^{\alpha_0-1}e^{\kappa_0\lambda} \times \lambda^t e^{-n\lambda} \propto \lambda^{\alpha_0 + t - 1}e^{-(\kappa_0 + n)\lambda},$$

where we recognize the right-hand side as the kernel of $\mathsf{Gamma}(\alpha_n = \alpha_0 + t, \kappa_n = \kappa_0 + n).$

Interpretation and probability interval estimates. In this framework, suppose we observed no events ($t = 0$) in $n = 100$ experiments. Then $\alpha_n = \alpha_0 = 0.002$ and $\kappa_n = \kappa_0 + n = 100.001.$ Thus $P(\lambda < .001) = 0.9964,$ so we are even more skeptical than before that such events occur in real life.

pgamma(.001, .002, 100.001)
## 0.9963579

By contrast, suppose we observed $t = 3$ events in $n = 100$ experiments. Then $\alpha_n = 3.002,\,$ $\kappa_n = 100.001$ and $P(\lambda < .001) = .00015,$ so we no longer believe such events are impossible.

pgamma(.001, 3.002, 100.001)
## 0.0001535587

It is easy to investigate the influence of the prior distribution on the resulting posterior probability interval. Try several different 'reasonably skeptical' priors and you will find that the data overwhelm the priors to give a result that is not a lot different from the one shown above.

An advantage of the Bayesian framework is that it lends itself to iterative experimentation: The posterior distribution from one series of experiments can serve as the prior distribution for interpreting future experiments.

On one hand, if we continue to see isolated occurrences of our event in future experiments, then we will find interval estimates that include values ever more markedly separated from $0$. On the other hand, if repeated future experiments fail to reveal any additional occurrences, we may eventually get posterior probability intervals that provide grounds to believe that the three occurrences recorded earlier might have resulted from experimental error.

---

Notes: (1) You may not become an 'instant fan' of the Bayesian approach. But it treats your issue seriously based on experimentation at hand. If you agree with the prior distribution then logically you must agree with the consequences of the posterior distribution.

(2) Acknowledgment: The development in this Answer is somewhat similar in notation and approach to a quite different gamma-Poisson Bayesian example in Chapter 8 of Suess (2010).