Mutually exclusive events

Under the four-form pronoun representation, one observed token is either nominative or accusative. A single form cannot have both surface cases, regardless of the probabilities assigned to the forms.

Let

N\equiv\{\textit{he},\textit{they}\}

be the nominative event and let

A\equiv\{\textit{him},\textit{them}\}

be the accusative event. The two sets share no outcomes:

N\cap A=\varnothing.

Events whose intersection is empty are mutually exclusive. They cannot both occur on one realization of the outcome represented by \Omega.

The definition concerns set membership:

B\text{ and }C\text{ are mutually exclusive} \quad\Longleftrightarrow\quad B\cap C=\varnothing.

The definition does not mention probability. We can determine whether two events are mutually exclusive before assigning probabilities to them.

Additivity for mutually exclusive events

Under the toy probability measure,

\mathbb{P}(N)=.20+.10=.30

and

\mathbb{P}(A)=.30+.40=.70.

Since N and A are mutually exclusive, their union contains two nonoverlapping parts. The probability measure thus gives

\begin{aligned} \mathbb{P}(N\cup A) &=\mathbb{P}(N)+\mathbb{P}(A)\\ &=.30+.70\\ &=1. \end{aligned}

In this representation, N\cup A=\Omega_4. The sum equals one because the two events are both exclusive and exhaustive.

Mutual exclusivity and exhaustivity are distinct properties. The singleton events \{\textit{he}\} and \{\textit{them}\} are mutually exclusive because they share no outcome. They are not exhaustive because their union is not \Omega_4. Their probabilities sum to .20+.40=.60, not to one.

A nonexclusive pair

Let

P\equiv\{\textit{they},\textit{them}\}

be the plural event. Compare P with the accusative event A. Their intersection is

P\cap A=\{\textit{them}\}.

The pair is not mutually exclusive because them belongs to both events. Directly adding \mathbb{P}(P)=.50 and \mathbb{P}(A)=.70 would count the probability assigned to them twice, producing the invalid value 1.20.

Check whether the sets overlap before adding their probabilities.

Code

nominative <- c("he", "they")
accusative <- c("him", "them")
plural <- c("they", "them")

length(intersect(nominative, accusative)) == 0
length(intersect(plural, accusative)) == 0

The first expression returns TRUE; the second returns FALSE.

Dependence on the represented outcome

Mutual exclusivity is relative to the outcome represented by one realization of the model. If one outcome is one pronoun token, nominative and accusative forms are mutually exclusive in the toy representation. If one outcome is a sentence containing two pronouns, the events “the sentence contains a nominative pronoun” and “the sentence contains an accusative pronoun” may occur together.

The labels nominative and accusative are the same in both examples. The intersections differ because an outcome represents one token in the first example and one sentence in the second.

Labels do not guarantee mutual exclusivity

Contrasting linguistic labels do not by themselves imply an empty intersection. Terms such as singular and plural often denote mutually exclusive values of one annotation. But terms such as plural and accusative denote values on different dimensions and may overlap.

Write the events as sets and compute their intersection. If the intersection is empty, the events are mutually exclusive under the declared sample space. The linguistic labels alone cannot establish this relation.

Check your understanding

Are \{\textit{he},\textit{him}\} and \{\textit{they},\textit{them}\} mutually exclusive? Are they exhaustive?
Let F be the first person event and P the plural event in the larger pronoun sample space. Write F\cap P and decide whether the events are mutually exclusive.
Give two mutually exclusive events that are not exhaustive in \Omega_4.
Explain why changing the outcome from one token to one sentence can change whether two linguistic events are mutually exclusive.

--- title: "Mutually exclusive events" --- Under the four-form pronoun representation, one observed token is either nominative or accusative. A single form cannot have both surface cases, regardless of the probabilities assigned to the forms. Let $$ N\equiv\{\textit{he},\textit{they}\} $$ be the nominative event and let $$ A\equiv\{\textit{him},\textit{them}\} $$ be the accusative event. The two sets share no outcomes: $$ N\cap A=\varnothing. $$ Events whose intersection is empty are [**mutually exclusive**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html). They cannot both occur on one realization of the outcome represented by $\Omega$. The definition concerns set membership: $$ B\text{ and }C\text{ are mutually exclusive} \quad\Longleftrightarrow\quad B\cap C=\varnothing. $$ The definition does not mention probability. We can determine whether two events are mutually exclusive before assigning probabilities to them. ## Additivity for mutually exclusive events Under the toy probability measure, $$ \mathbb{P}(N)=.20+.10=.30 $$ and $$ \mathbb{P}(A)=.30+.40=.70. $$ Since $N$ and $A$ are mutually exclusive, their union contains two nonoverlapping parts. The probability measure thus gives $$ \begin{aligned} \mathbb{P}(N\cup A) &=\mathbb{P}(N)+\mathbb{P}(A)\\ &=.30+.70\\ &=1. \end{aligned} $$ In this representation, $N\cup A=\Omega_4$. The sum equals one because the two events are both exclusive and exhaustive. Mutual exclusivity and exhaustivity are distinct properties. The singleton events $\{\textit{he}\}$ and $\{\textit{them}\}$ are mutually exclusive because they share no outcome. They are not exhaustive because their union is not $\Omega_4$. Their probabilities sum to $.20+.40=.60$, not to one. ## A nonexclusive pair Let $$ P\equiv\{\textit{they},\textit{them}\} $$ be the plural event. Compare $P$ with the accusative event $A$. Their intersection is $$ P\cap A=\{\textit{them}\}. $$ The pair is not mutually exclusive because *them* belongs to both events. Directly adding $\mathbb{P}(P)=.50$ and $\mathbb{P}(A)=.70$ would count the probability assigned to *them* twice, producing the invalid value $1.20$. Check whether the sets overlap before adding their probabilities. ```{r} #| eval: false nominative <- c("he", "they") accusative <- c("him", "them") plural <- c("they", "them") length(intersect(nominative, accusative)) == 0 length(intersect(plural, accusative)) == 0 ``` The first expression returns `TRUE`; the second returns `FALSE`. ## Dependence on the represented outcome Mutual exclusivity is relative to the outcome represented by one realization of the model. If one outcome is one pronoun token, nominative and accusative forms are mutually exclusive in the toy representation. If one outcome is a sentence containing two pronouns, the events “the sentence contains a nominative pronoun” and “the sentence contains an accusative pronoun” may occur together. The labels *nominative* and *accusative* are the same in both examples. The intersections differ because an outcome represents one token in the first example and one sentence in the second. ## Labels do not guarantee mutual exclusivity Contrasting linguistic labels do not by themselves imply an empty intersection. Terms such as *singular* and *plural* often denote mutually exclusive values of one annotation. But terms such as *plural* and *accusative* denote values on different dimensions and may overlap. Write the events as sets and compute their intersection. If the intersection is empty, the events are mutually exclusive under the declared sample space. The linguistic labels alone cannot establish this relation. ## Check your understanding 1. Are $\{\textit{he},\textit{him}\}$ and $\{\textit{they},\textit{them}\}$ mutually exclusive? Are they exhaustive? 2. Let $F$ be the first person event and $P$ the plural event in the larger pronoun sample space. Write $F\cap P$ and decide whether the events are mutually exclusive. 3. Give two mutually exclusive events that are not exhaustive in $\Omega_4$. 4. Explain why changing the outcome from one token to one sentence can change whether two linguistic events are mutually exclusive.