Foundational Concepts: Probability and Statistics

To grasp Bell's The­o­rem, a basic un­der­stand­ing of prob­a­bil­i­ty and sta­tis­tics is cru­cial.

Prob­a­bil­i­ty Basics

Prob­a­bil­i­ty quan­ti­fies the like­li­hood of an event oc­cur­ring. For an event $A$, its prob­a­bil­i­ty $P(A)$ ranges from 0 (im­pos­si­ble) to 1 (cer­tain).

• Mu­tu­al­ly Ex­clu­sive Events: If events $A$ and $B$ cannot occur si­mul­ta­ne­ous­ly, the prob­a­bil­i­ty of either $A$ or $B$ oc­cur­ring is $P(A \cup B) = P(A) + P(B)$.
• In­de­pen­dent Events: If the oc­cur­rence of event $A$ does not affect the prob­a­bil­i­ty of event $B$, then $P(A \cap B) = P(A)P(B)$.

Ex­pec­ta­tion Value

The ex­pec­ta­tion value (or ex­pect­ed value) of a random vari­able $X$, de­not­ed $E[X]$ or $\langle X \rangle$, is the weight­ed av­er­age of all pos­si­ble values that $X$ can take, with weights being their re­spec­tive prob­a­bil­i­ties.

If $X$ can take values $x_1, x_2, \ldots, x_n$ with prob­a­bil­i­ties $P(x_1), P(x_2), \ldots, P(x_n)$, then:

For ex­am­ple, if a mea­sure­ment can yield out­comes $+1$ or $-1$, each with prob­a­bil­i­ty $1/2$, the ex­pec­ta­tion value is $E[X] = (+1) \cdot (1/2) + (-1) \cdot (1/2) = 0$.

Cor­re­la­tion Co­ef­fi­cient

A cor­re­la­tion co­ef­fi­cient mea­sures the sta­tis­ti­cal re­la­tion­ship, or as­so­ci­a­tion, be­tween two random vari­ables. For two vari­ables $A$ and $B$, the cor­re­la­tion co­ef­fi­cient $C(A, B)$ (often de­not­ed $\rho_{AB}$) typ­i­cal­ly ranges from $-1$ to $+1$.

• $C(A, B) = +1$: Per­fect pos­i­tive cor­re­la­tion ( $A$ and $B$ always agree).
• $C(A, B) = -1$: Per­fect neg­a­tive cor­re­la­tion ( $A$ and $B$ always dis­agree).
• $C(A, B) = 0$: No linear cor­re­la­tion.

The ex­pec­ta­tion value of the prod­uct of two vari­ables is often used to define cor­re­la­tion in the con­text of Bell's the­o­rem. If $A$ and $B$ are vari­ables that can take values $\pm 1$, their cor­re­la­tion is given by:

Where $P(i, j|A,B)$ is the joint prob­a­bil­i­ty of ob­serv­ing out­come $i$ for $A$ and out­come $j$ for $B$.