Probability

{{bronze}}{{mathematics}} 
'''Probability''' is a non-negative additive set function whose maximum value is unity. Put more simply, probability is a function that assigns to an event a real number in the interval [0,1] inclusive.{{efn|Zero corresponds to false or cannot occur, and one corresponds to true or must occur.}} The formal study of probability is a mathematical discipline known as probability theory. The applications of probability theory are useful for the analysis of [[random]] events (''i.e.'', events that occur in no discernible pattern) and the assessment of the relationship between [[evidence]] and [[hypotheses]], or [[belief]]s more generally, amongst other things.

==Interpretations of probability==
'''Classical probability''' assumes a uniform probability distribution. In the discrete case, this is represented by a ''random variable'' representing some event with a countable number of possible outcomes, and the probability of all outcomes equal:

:<math>P(X = x) = \frac{1}{\left|\omega\right|}</math>

where <math>\omega</math> is the set of possible outcomes.

This can be illustrated with packs of cards. With 52 cards and 13 of each suit, the odds of drawing a specific suit, (''e.g.'', clubs) is 1 in 4, as 1 in 4 of the available cards are clubs.

A common error with classical probability is to use over-simplified statistical models that do not take all factors into account and groundlessly assume that all outcomes are equally probable; for example, assuming a 50-50 chance of heads when flipping a loaded coin, or assuming a 1-in-4 probability of drawing a card of a certain suit from a trick deck, or assuming that since there are four possible gender-pairings (man-man, man-woman, woman-man, and woman-woman),{{efn|This is a false premise, but that's another story: [[non-binary gender]].}} half of all people are [[gay]].<ref name="Swartz">http://www.sfu.ca/~swartz/obdurate.htm</ref>

Misconceptions such as the [[balance fallacy]] occur when people treat such unequal outcomes as equally probable; as silly as it sounds when it is actually explained, this mistake is still quite common. [[Philosophy]] professor Norman Swartz links this [[Genuine but insignificant cause|oversimplifying analysis]] to [[Rene Descartes|Cartesian]] cogito-ergo-sum rationalism, which completely rejects [[empirical]] observation as a "way of knowing."<ref name="Swartz"/>

With '''frequentist probability''', the probability is ''measured'', rather than predicted, so is useful in a range of [[scientific]] disciplines. It also overcomes the issue of having to predict unequal outcomes by looking at empirical data. This does require the (entirely reasonable, if not blatantly obvious) assumption that the probability of an event is related to the frequency with which it occurs.

An important distinction is between frequentist and '''[[Bayesian]] probability'''. Frequentist probability is based entirely on repeatable events, and how frequently they occur; for example, we can determine the probability of earthquakes by studying seismic records. Bayesian probability, by contrast, can also assign ''prior probabilities'' to hypotheses (which in frequentism are binary propositions); these prior probabilities can then be tested against observed data to produce a ''posterior probability'' taking the data into account. This allows for the empirical refinement of probability models.

In its most basic form, it is the measure of confidence, or belief, that a person holds in a proposition. Using Bayesian probability allows a researcher to judge the amount of confidence that they have in a particular result.

==Misuse by creationists==
{{main|1=Argument from fine tuning|3=Creationism}}
Many <strike>nut jobs</strike> [[creation science|creation scientists]] love to mention how so many of the things we observe around us have a probability of occurring of nearly [[zero]]. They love that "nearly zero" thing. They seem to believe that if the odds are low enough, then what we see couldn't have happened randomly. However, since we do observe the "whatever improbable thing", then it must have made it despite the odds.<ref>See the discussion in {{cite book|title=The Failures of Mathematical Anti-Evolutionism|author=Jason Rosenhouse|date=2022|publisher=Cambridge U. Press|isbn=9781108820448|chapter=chapter 5 Probability Theory|pages=110-162}}</ref>

Again, they always mention the "nearly zero" thing. Actually, they mean zero. They pretend to give the observation of the low probability that it might have, but they truly believe that it had ''zero'' probability.  That is perfectly reasonable given that their solution is that the observed effect was created [[supernatural]]ly by one or more supernatural entities.

In using probability to support their position, the creationists overlook two major points. Firstly, the ''prior probability'' of the [[Universe]] forming by chance (<math>P(\mathit{Chance})</math>) is not the same as the ''posterior probability'' of the universe forming by chance ''given that the universe has formed'' (<math>P(\mathit{Chance}|\mathit{Formed}))</math>. Secondly, even if there is a very low probability of an event occurring at some particular point, there is also a very low probability that that event will occur at ''none'' of the very many points in the universe (''i.e.'', the large size of the universe implies that there is a reasonable probability of there being an appropriate "[[Goldilocks Zone]]" ''somewhere'' in it).

Note furthermore that "nearly zero" probability can combine nicely with [[Teleology|teleological thinking]]. But if the chances of "random" [[molecule]]s making the leap to (say) something dauntingly [[irreducibly complex]] like a functioning living [[cell]] (as helpfully calculated by [[creationist mathematics|creationist arithmetic]]) seem vanishingly small, that's not the point — does it follow that the chances of somehow forming a proto-cell<ref>{{wpa|Protocell}}</ref> are equally vanishingly small? "[[Missing link]]s", anyone? (We <u>are</u> allowed to postulate [[missing links]], aren't we? — Those weird [[satanic]] principles of [[natural selection]] can really muck up the math of "nearly zero"…) 

===A hypothetical example===
As an example of the fallacy of looking at results and conjecturing backwards on what the probability of such occurring was, grab a set of playing cards and deal them out face up, one at a time. When the first card is dealt, the probability of it being ''whatever it is'', is 52<sup>-1</sup>, the probability of the second card being ''whatever it is'' is then 51<sup>-1</sup>, giving a combined probability of 2652<sup>-1</sup>.  Once you have got through the deck, the probability of you having dealt those cards ''in that order'' is a staggering one in 8×10<sup>67</sup>, which is pretty much "nearly zero", yet you just managed to do it! Congratulations, you're ''magic'' — just like [[Jesus]]!<ref group=note>With apologies to Sarah Silverman.</ref>

===What is the probability of the alternative?===
Let us assume that such-and-such is exceedingly improbable to have happened as a result of natural causes. 

Now let's compare that with the probability that it happened as a result of more-than-natural causes. As is typical with "intelligent design" advocates, we won't make any assumptions about the limitations on the designers, their methods, materials, or motives. They are apt to do anything.

Let's keep with the example of playing cards. With unlimited and inscrutable designers, they are not restricted to using A through K of clubs, diamonds, hearts and spades. Of course, they can use as many jokers as they want, but also they could add in a deck of tarot cards, Uno cards or filing cards. Given that range of possibilities, what is the probability that they would choose to end up with only cards from a standard deck? What is the probability that an intelligent designer aiming to create a [[DNA]] sequence would choose to use only [[hydrogen]], [[carbon]], [[oxygen]], nitrogen and so on, rather than also using [[helium]], plutonium, neutronium, and anti-hydrogen? If the probability (for both of these) isn't '''0''', it's certainly a '''lot''' less than any creationist has dreamed up about evolution.

==Zero probability==
:''See also: [[Improbable things happen]]''
 
There is a difference between an event that "will never occur" and one that "has a zero probability of occurring."

*To say that something "will never occur" means there is no option for it to occur (''i.e.'', it is an outcome outside the random variable's set <math>\omega</math> of outcomes).  For example, flip a fair coin with a thickness of zero and the event that "neither heads nor tails is flipped" ''will never occur'', since the coin only has heads or tails.
*To say that something "has a zero probability of occurring" means that its probability is statistically equivalent to zero. This is seen in continuous random variables where the number of possible outcomes is uncountable. For example, if a random variable <math>X</math> could have as an outcome any real number between -1 and 1, the probability of any individual real-number outcome (''e.g.'', <math>P(X = 0)</math>) would be an infinitesimal quantity (<math>\left|\mathbb{R}\right|^{-1}</math>), which is statistically equivalent to zero. However, the probability of the outcome falling within some ''range'' of the possibilities (''e.g.'', <math>P(X \leq 0)</math>), may be nonzero. This is analogous to the computation of integrals, where a non-zero area is calculated by summing infinitesimal slivers of it. If infinitesimal slivers are too abstract, this means that, probabilistically, the chances of, for example, your weight being exactly 152.8 pounds is zero (because the "sliver" has zero width), but the probability of you being between 152 and 153 pounds is somewhat greater than zero (the sliver has actual width).{{efn|For those who don't live in the US, your probability may still be zero because ... [[metric system|modernity]].}}
==See also==
*[[Borel's Law]]
*[[Fuzzy logic]]
*[[Holmesian fallacy]]

==External links==
* [http://students.brown.edu/seeing-theory/basic-probability/index.html Seeing Theory: Basic Probability]
* [http://world.mathigon.org/Probability Understanding Probability]

==Notes==
{{reflist|group=note|2}}

==References==
{{reflist}}

[[Category:Mathematics]]

Categories: Mathematics