The categorical distribution permits any finite support. What changes when the scientific question divides those outcomes into just two classes, target and nontarget? The Bernoulli family represents that binary case.
Return to the case-coded pronoun sample space and define the accusative event
A_{14}\equiv\{\textit{me},\textit{you}_{[+\mathrm{acc}]},\textit{them}, \textit{her},\textit{him},\textit{it}_{[+\mathrm{acc}]},\textit{us}\}.
Define the indicator random variable
X(\omega) \equiv \begin{cases} 1,&\omega\in A_{14},\\ 0,&\omega\notin A_{14}. \end{cases}
The range of X is \{0,1\} even though the sample space contains fourteen pronoun outcomes. This is the decisive compression: the Bernoulli restriction concerns the random variable’s range, not the size of \Omega.
The binary code presupposes a case annotation for syncretic surface forms such as you and it. The Universal Dependencies English guidelines use syntactic context for that annotation; the surface string alone does not supply it.
Basically, one parameter records the probability of the event coded as 1, and the remaining probability goes to 0. More specifically, the Bernoulli distribution assigns probability to a binary random variable. For \pi\in[0,1], we write
X\sim\operatorname{Bernoulli}(\pi)
when
p_X(x) \equiv\mathbb P(X=x) =\pi^x(1-\pi)^{1-x}, \qquad x\in\{0,1\}.
Substituting the two support values gives
p_X(1)=\pi \qquad\text{and}\qquad p_X(0)=1-\pi.
Thus \pi is the probability of the event coded as one. The definition of X makes that event explicit.
Set \pi=.27. Then
p_X(1)=.27 \qquad\text{and}\qquad p_X(0)=.73.
case_code <- 0:1
case_mass <- dbinom(case_code, size = 1, prob = .27)
plot(case_code, case_mass, type = "h", lwd = 5, lend = 1,
col = "#B2182B", xaxt = "n", xlab = "Case", ylab = "Probability",
main = "PMF for the Bernoulli distribution on pronoun case",
ylim = c(0, .8), bty = "l")
axis(1, at = case_code, labels = c("[-acc]", "[+acc]"))
points(case_code, case_mass, pch = 19, col = "#B2182B")
The expected value is
\begin{aligned} \mathbb E[X] &=0(1-\pi)+1(\pi)\\ &=\pi. \end{aligned}
The variance is
\begin{aligned} \operatorname{Var}(X) &=(0-\pi)^2(1-\pi)+(1-\pi)^2\pi\\ &=\pi(1-\pi). \end{aligned}
For \pi=.27, \operatorname{Var}(X)=.27(.73)=.1971.
If the surface forms you and it are not split into case-coded outcomes, a binary case variable is not defined for them without an additional annotation rule. Let A_{12} and N_{12} be the overlapping accusative and nonaccusative events in the twelve-form space. One option is a three-valued variable:
C(\omega) \equiv \begin{cases} 2,&\omega\in A_{12}\cap N_{12},\\ 1,&\omega\in A_{12}\setminus N_{12},\\ 0,&\omega\in N_{12}\setminus A_{12}, \end{cases}
This variable is categorical, not Bernoulli. Alternatively, a pair of indicators can record membership in A_{12} and N_{12} separately.