Does it make sense when people say "statistically impossible"?
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$\begingroup$ Suppose you are flipping a coin. A statistically impossible event would be to flip both heads and tails. Though, I'm not sure this answers your question. Could you detail what you mean by "statistically impossible"? $\endgroup$– user141854Dec 8, 2016 at 15:31
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$\begingroup$ To me, the phrase means, informally, that the probability of observing $X$ naturally is so low that fraud, trickery, error, or some such thing is more likely. If my poker opponent gets dealt two royal flushes in a row, for example, I don't think I'd assume that this was just his lucky day. $\endgroup$– luluDec 8, 2016 at 15:32
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$\begingroup$ in my neighborhood, everyone is a genius. $\endgroup$– user25406Dec 8, 2016 at 15:34
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1$\begingroup$ 130,000 votes being in favor of a single presidential candidate. $\endgroup$– builder-7000Nov 7, 2020 at 7:07
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$\begingroup$ @user141854 That's logically impossible, not statistically impossible. As for your request, that's precisely what the OP is asking for. $\endgroup$– Jim BalterDec 20, 2022 at 9:14
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4 Answers
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A statistical impossibility is a probability that is so low as to not be worthy of mentioning. Sometimes it is quoted as $10^{-50}$ although the cutoff is inherently arbitrary. Although not truly impossible the probability is low enough so as to not bear mention in a rational, reasonable argument.
In some cases that arise in Gedanken experiments in thermodynamics, the probabilities can be approximately $10^{- \textrm{Avogadro's number}}$, that is, $10^{-10^{23}}$, give or take a few billion orders of magnitude.The standard framework of probability theory attempts to assign to each outcome $X$ a number $P(X)$ between $0$ and $1$, which we call the probability of the outcome. The higher the number, the more likely that outcome is to occur. Depending on context, a sufficiently small value of $P(X)$ will correspond to something being improbable, but there's no inherent threshold between improbable and likely.
General probability theory does not have a good way of making sense of "impossible" or "necessity". Having a probability of $0$ does not mean something cannot happen, and having a probability of $1$ does not mean that something must happen. To explain this with a metaphor from geometry, consider the notion of area. The area of nothing is $0$, but so is the area of a single point. In that sense, the notion of area cannot distinguish between nothing and that which is "infinitely small".
Similarly, when trying to applying probability theory to problems that have infinitely many outcomes, the probability of an event can be zero even if it is possible, so long as there are infinitely many other possibilities that are just as likely (or some similar situation). When it comes down to it, the real number line does not have infinitely small numbers, and since probability theory uses real numbers, these events can only be assigned a probability of $0$.
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1$\begingroup$ Shouldn't there be a negative sign in the exponent of $10^{-10^{23}}$? $\endgroup$– mweissDec 8, 2016 at 15:31
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1$\begingroup$ In some circumstances (mathematical models) things that have probability zero are nevertheless not impossible as Rohan's answer points out. In some other situations however probability zero and and being impossible DO coincide, like when throwing dice. The probability of throwing 7 with an ordinary die is 0, and the event is indeed impossible. (I'm another Vincent than the one answering below) $\endgroup$– VincentDec 19, 2016 at 15:09
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$\begingroup$ @Vincent, could you provide a similar example of an outcome that is not impossible but has a probability of zero? $\endgroup$– SmacLFeb 9, 2017 at 8:06
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1$\begingroup$ @ShaneMacLaughlin The answer by the other Vincent does provide such an example: pick a number in [0,1] from the uniform distribution. Then the probability of picking 1/2 is zero, but it might just happen. $\endgroup$– VincentFeb 9, 2017 at 8:52
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$\begingroup$ In practical use, "statistically impossible" is dishonest rhetoric used to cast doubt on an actual event, as when Trump claims that Biden's win was "statistically impossible". Serious people should not use the term as it has no legitimate mathematical meaning. $\endgroup$ Dec 20, 2022 at 9:18
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In statistics there is nothing impossible, you may want to change this to improbable which is something completely different. Theoretically speaking everything could have 1÷∞ = ~0 probability but that just not mean it could actually happen at least in our life cycle. In addition some of the things that we consider more sure to happen just because they gather a large percentage of probability don't always appear to follow that rule.
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$\begingroup$ Right. Other answers are trying to rationalize this bogus term. In practical use, "statistically impossible" is dishonest rhetoric used to cast doubt on an actual event, as when Trump claims that Biden's win was "statistically impossible". Serious people should not use the term as it has no legitimate mathematical meaning. $\endgroup$ Dec 20, 2022 at 9:28
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It can mathematically make sense. You look like you are thinking about discrete probabilities. In continuous probabilities, you define what is called a density function, and whenever it is finite in some value $x$, the probability of picking $x$ is null.
If you are not familiar with the concept, consider $[0,1]$. Give each number in this interval an equal probability of being picked. The probability of picking $0.5$ is null. However, the probability of picking a number smaller or equal to $0.2$ is $0.2$.
It does not make sense in a finite world, though. Using continuous mathematics to describe it do not change that.
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Consider the exponential equation, which never touches the value 0, but at what point would it be considered impossible or improbable to occur? This equation describes radioactive decay and so at what point can one safely say there is no more radiation coming from the material? IF you knew the exact number of atoms, you might be able to reach a point of saying that the last unstable atom had moved to a stable state...but usually it is considered safe after only 10 half-lives. But that is just a practical application and not the area that truly approaches zero.
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$\begingroup$ Exponential functions are deterministic, not something random characterized by statistics. $\endgroup$– ЅᴀᴀᴅJun 18, 2018 at 14:46
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$\begingroup$ @SAAD Statistics doesn't just apply to "something random" ... whether something is "deterministic" isn't relevant. Statistics comes into play because we don't have perfect knowledge. $\endgroup$ Dec 20, 2022 at 9:30