Probability

Prob­a­bil­ities are the cen­tral sub­ject of the dis­ci­pline of Prob­a­bil­ity the­ory. \(\mathbb{P}(X)\) de­notes our level of be­lief, or some­one’s level of be­lief, that the propo­si­tion \(X\) is true. In the clas­si­cal and canon­i­cal rep­re­sen­ta­tion of prob­a­bil­ity, 0 ex­presses ab­solute in­cre­dulity, and 1 ex­presses ab­solute cre­dulity.

knows-req­ui­site(Abil­ity to read logic): Fur­ther­more, mu­tu­ally ex­clu­sive events have ad­di­tive clas­si­cal prob­a­bil­ities: \(\mathbb{P}(X \wedge Y) = 0 \implies \mathbb{P}(X \vee Y) = \mathbb{P}(X) + \mathbb{P}(Y).\)

For the stan­dard prob­a­bil­ity ax­ioms, see https://​​en.wikipe­dia.org/​​wiki/​​Prob­a­bil­ity_ax­ioms. write up page on ar­bital about prob­a­bil­ity ax­ioms.

Notation

\(\mathbb{P}(X)\) is the prob­a­bil­ity that X is true.

\(\mathbb{P}(\neg X) = 1 - \mathbb{P}(X)\) is the prob­a­bil­ity that X is false.

\(\mathbb{P}(X \wedge Y)\) is the prob­a­bil­ity that both X and Y are true.

\(\mathbb{P}(X \vee Y)\) is the prob­a­bil­ity that X or Y or both are true.

\(\mathbb{P}(X|Y) := \frac{\mathbb{P}(X \wedge Y}{\mathbb{P}(Y)}\) is the con­di­tional prob­a­bil­ity of X given Y. That is, \(\mathbb{P}(X|Y)\) is the de­gree to which we would be­lieve X, as­sum­ing Y to be true. \(\mathbb{P}(yellow|banana)\) ex­presses “The prob­a­bil­ity that a ba­nana is yel­low.” \(\mathbb{P}(banana|yellow)\) ex­presses “The prob­a­bil­ity that a yel­low thing is a ba­nana”.

Cen­tral­ity of the clas­si­cal representation

While there are other ways of ex­press­ing quan­ti­ta­tive de­grees of be­lief, such as odds ra­tios, there are sev­eral es­pe­cially use­ful prop­er­ties or roles of clas­si­cal prob­a­bil­ities that give them a cen­tral /​ con­ver­gent /​ canon­i­cal sta­tus among pos­si­ble ways of rep­re­sent­ing cre­dence.

Odds ra­tios are iso­mor­phic to prob­a­bil­ities—we can read­ily go back and forth be­tween a prob­a­bil­ity of 20%, and odds of 1:4. But un­like odds ra­tios, prob­a­bil­ities have the fur­ther ap­peal­ing prop­erty of be­ing able to add the prob­a­bil­ities of two mu­tu­ally ex­clu­sive pos­si­bil­ities to ar­rive at the prob­a­bil­ity that one of them oc­curs. The 16 prob­a­bil­ity of a six-sided die turn­ing up 1, plus the 16 prob­a­bil­ity of a die turn­ing up 2, equals the 13 prob­a­bil­ity that the die turns up 1 or 2. The odds ra­tios 1:5, 1:5, and 1:2 don’t have this di­rect re­la­tion (though we could con­vert to prob­a­bil­ities, add, and then con­vert back to odds ra­tios).

Thus, clas­si­cal prob­a­bil­ities are uniquely the quan­tities that must ap­pear in the ex­pected util­ities to weigh how much we pro­por­tion­ally care about the un­cer­tain con­se­quences of our de­ci­sions. When an out­come has clas­si­cal prob­a­bil­ity 13, we mul­ti­ply the de­gree to which we care by a fac­tor of 13, not by, e.g., the odds ra­tio 1:2.

If the amount you’d pay for a lot­tery ticket that paid out on 1 or 2 was more or less than twice the price you paid for a lot­tery ticket that only paid out on 1, or a lot­tery ticket that paid out on 2, then I could buy from you and sell to you a com­bi­na­tion of lot­tery tick­ets such that you would end up with a cer­tain loss. This is an ex­am­ple of a Dutch book ar­gu­ment, which is one kind of co­her­ence the­o­rem that un­der­pins clas­si­cal prob­a­bil­ity and its role in choice. (If we were deal­ing with ac­tual bet­ting and gam­bling, you might re­ply that you’d just re­fuse to bet on dis­ad­van­ta­geous com­bi­na­tions; but in the much larger gam­ble that is life, “do­ing noth­ing” is just one more choice with an un­cer­tain, prob­a­bil­is­tic pay­off.)

The com­bi­na­tion of sev­eral such co­her­ence the­o­rems, most no­tably in­clud­ing the Dutch Book ar­gu­ments, Cox’s The­o­rem and its vari­a­tions for prob­a­bil­ity the­ory, and the Von Neu­mann-Mor­gen­stern the­o­rem (VNM) and its vari­a­tions for ex­pected util­ity, to­gether give the clas­si­cal prob­a­bil­ities be­tween 0 and 1 a cen­tral sta­tus in the the­ory of epistemic and in­stru­men­tal ra­tio­nal­ity. Other ways of rep­re­sent­ing scalar prob­a­bil­ities, or al­ter­na­tives to scalar prob­a­bil­ity, would need to be con­verted or munged back into clas­si­cal prob­a­bil­ities in or­der to an­i­mate agents mak­ing co­her­ent choices.

This also sug­gests that bounded agents which ap­prox­i­mate co­her­ence, or at least man­age to avoid blatantly self-de­struc­tive vi­o­la­tions of co­her­ence, might have in­ter­nal men­tal states which can be ap­prox­i­mately viewed as cor­re­spond­ing to clas­si­cal prob­a­bil­ities. Per­haps not in terms of such agents nec­es­sar­ily con­tain­ing float­ing-point num­bers that di­rectly rep­re­sent those prob­a­bil­ities in­ter­nally, but at least in terms of our be­ing able to look over the agent’s be­hav­ior and de­duce that they were “be­hav­ing as if” they had as­signed some co­her­ent clas­si­cal prob­a­bil­ity.

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  • Probability theory

    The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.