= Joint probability of empty and strictly positive detector-width intervals
{title2=$(N_E,N_P,N_Z)\sim\operatorname{Multinomial}(N;p_E,p_P,p_Z)$}
The \b[joint probability of empty and strictly positive detector-width intervals] must distinguish outcomes of one measurement from counts across repeated measurements. For one normal <variance>-difference estimate $\widehat D\sim N(D,s^2)$, let $I=[\widehat D-zs,\widehat D+zs]$. An empty physical width set requires $\widehat D+zs<0$; a strictly positive interval requires $\widehat D-zs>0$. Those events are disjoint for that same interval, so their intersection has probability zero.
Writing $d=D/s$ and $\Phi$ for the standard normal cumulative distribution function, the three category probabilities are
$$
p_E=\Phi(-z-d),\qquad
p_P=\Phi(d-z),\qquad
p_Z=1-p_E-p_P.
$$
The last category is an interval containing zero. At $D=0$ and a central $1-\alpha$ interval, these become $(\alpha/2,\alpha/2,1-\alpha)$, as in <boundary probability of an empty detector-width confidence set>. For interior $D>0$, the categories describe the relation to zero; they are not themselves the coverage categories for the true positive value.
For $N$ independent repetitions with the same probabilities, the counts $(N_E,N_P,N_Z)$ have a <multinomial distribution>. For nonnegative integers $e,p,k$ summing to $N$,
$$
\Pr(N_E=e,N_P=p,N_Z=k)=\frac{N!}{e!p!k!}p_E^ep_P^pp_Z^k.
$$
Although the measurement repetitions are independent, their category counts are not. An empty and a positive indicator cannot both equal one in the same repetition, giving
$$
\operatorname{Cov}(N_E,N_P)=-Np_Ep_P.
$$
This <negative covariance of empty and positive detector-interval counts> follows from the fixed total count.
For an original example, take 20 independent boundary measurements with exact central 95 percent normal intervals. Then $p_E=p_P=0.025$. Each count has mean $0.5$, and their <covariance> is $-0.0125$. The probability of exactly one empty and one strictly positive interval is $20\times19\times0.025^2\times0.95^{18}$, about $0.094$. It is not zero, unlike the intersection of those two outcomes for a single interval.
NIST's uncertainty guidance emphasizes the assumptions needed to assign confidence levels.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§§6.2–6.3, Expanded Uncertainty]{ref} The joint-count calculation above additionally assumes independent repetitions with a common normal model; shared calibration errors can invalidate that independence even when each marginal interval is computed in the same way.
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