Mutually exclusive events

Under the four-form pronoun representation, one observed token is either nominative or accusative. Can we obtain the probability of either case by adding the two event probabilities? Yes, but only because a single form cannot have both surface cases. We first make that fact precise.

Let

N_4\equiv\{\textit{he},\textit{they}\}

be the nominative event and let

A_4\equiv\{\textit{him},\textit{them}\}

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

N_4\cap A_4=\varnothing.

Basically, two events are mutually exclusive when one outcome cannot make both happen. More specifically, 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_4)=.20+.10=.30

and

\mathbb{P}(A_4)=.30+.40=.70.

Since N_4 and A_4 are mutually exclusive, their union contains two disjoint parts. The probability measure thus gives

\begin{aligned} \mathbb{P}(N_4\cup A_4) &=\mathbb{P}(N_4)+\mathbb{P}(A_4)\\ &=.30+.70\\ &=1. \end{aligned}

In this representation, N_4\cup A_4=\Omega_4. The sum equals one because the two events are both disjoint 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, but they are not exhaustive because their union is not \Omega_4. Their probabilities thus sum to .20+.40=.60, not to one.

A nonexclusive pair

Now we can see why the empty-intersection check matters. If two events overlap, adding their probabilities creates a double-counting problem.

Let

P_4\equiv\{\textit{they},\textit{them}\}

be the plural event. Compare P_4 with the accusative event A_4. Their intersection is

P_4\cap A_4=\{\textit{them}\}.

The pair is not mutually exclusive because them belongs to both events. Directly adding \mathbb{P}(P_4)=.50 and \mathbb{P}(A_4)=.70 would count the probability assigned to them twice, producing the invalid value 1.20.

Check whether the sets overlap before adding their probabilities. This set-level check tells us whether the simple addition rule is licensed.

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.

The exact upshot is that mutual exclusivity is an empty-intersection claim, not a claim about labels or probability size. The next page assigns probability to intersections directly, including intersections that are not empty.

Check your understanding

Are \{\textit{he},\textit{him}\} and \{\textit{they},\textit{them}\} mutually exclusive? Are they exhaustive?
Let F_{12} be the first-person event and P_{12} the plural event in the twelve-form pronoun sample space. Write F_{12}\cap P_{12} 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. Can we obtain the probability of either case by adding the two event probabilities? Yes, but only because a single form cannot have both surface cases. We first make that fact precise. Let $$ N_4\equiv\{\textit{he},\textit{they}\} $$ be the nominative event and let $$ A_4\equiv\{\textit{him},\textit{them}\} $$ be the accusative event. The two sets share no outcomes: $$ N_4\cap A_4=\varnothing. $$ Basically, two events are mutually exclusive when one outcome cannot make both happen. More specifically, 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, $$ \mathbb{P}(N_4)=.20+.10=.30 $$ and $$ \mathbb{P}(A_4)=.30+.40=.70. $$ Since $N_4$ and $A_4$ are mutually exclusive, their union contains two disjoint parts. The probability measure thus gives $$ \begin{aligned} \mathbb{P}(N_4\cup A_4) &=\mathbb{P}(N_4)+\mathbb{P}(A_4)\\ &=.30+.70\\ &=1. \end{aligned} $$ In this representation, $N_4\cup A_4=\Omega_4$. The sum equals one because the two events are both disjoint 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, but they are not exhaustive because their union is not $\Omega_4$. Their probabilities thus sum to $.20+.40=.60$, not to one. ## A nonexclusive pair Now we can see why the empty-intersection check matters. If two events overlap, adding their probabilities creates a **double-counting problem**. Let $$ P_4\equiv\{\textit{they},\textit{them}\} $$ be the plural event. Compare $P_4$ with the accusative event $A_4$. Their intersection is $$ P_4\cap A_4=\{\textit{them}\}. $$ The pair is not mutually exclusive because *them* belongs to both events. Directly adding $\mathbb{P}(P_4)=.50$ and $\mathbb{P}(A_4)=.70$ would count the probability assigned to *them* twice, producing the invalid value $1.20$. Check whether the sets overlap before adding their probabilities. This set-level check tells us whether the simple addition rule is licensed. ```{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. The exact upshot is that mutual exclusivity is an empty-intersection claim, not a claim about labels or probability size. The [next page](joint-probability-of-events.qmd) assigns probability to intersections directly, including intersections that are not empty. ## Check your understanding 1. Are $\{\textit{he},\textit{him}\}$ and $\{\textit{they},\textit{them}\}$ mutually exclusive? Are they exhaustive? 2. Let $F_{12}$ be the first-person event and $P_{12}$ the plural event in the twelve-form pronoun sample space. Write $F_{12}\cap P_{12}$ 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.