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For a binomial distribution with $p_n\to 0$ and $n{p}_n\to \infty$, the Bennett inequality at $C$ standard deviations tends to $2e^{-C^2/2}$, whereas the Hoeffding inequality tends to two. The raw Bennett inequality bound is eventually smaller for any fixed $C>0$; after clipping at one it is nontrivial in the limit only when $C>\sqrt{2\log 2}$.

A Galton-Watson process whose offspring law is a binomial distribution $\mathrm{Bin}(n,p)$ has mean $np$ and the displayed offspring probability generating function. Its branching extinction probability is the smallest root of $q=f(q)$ in $[0,1]$. It dominates a breadth-first exploration of a binomial random graph by allowing each explored vertex up to $n$ independent possible children, including previously exposed or already discovered vertices as phantom children.

A chain-binomial epidemic model uses discrete time, with conditionally independent binomial distributions for infection and recovery counts given the current compartments. It requires valid probability parameters; replacing $1-e^{-\beta I\delta t}$ by $\beta I\delta t$ is a small-step approximation.

The Hamming cube is $\{-1,1{\}}^n$, or equivalently $\{0,1{\}}^n$, equipped with Hamming distance. Under its uniform probability measure, the distance from a fixed vertex has the binomial distribution with parameters $n,1/2$. This converts geometric ball sizes into probability tails.

A Lévy family of graphs consists of finite connected graphs with their uniform probability measures and graph distances divided by their diameters, whose concentration of measure functions vanish at every fixed positive normalized radius. Equivalently, every family occupying at least half the vertices has a radius-$\lfloor \epsilon D_n\rfloor$ neighbourhood occupying a proportion tending to one, uniformly over such families. Harper theorem and binomial distribution tail bounds prove this for the hypercube graphs.

A logistic-normal count model takes $Y_i\mid {\theta }_i\sim \mathrm{Bin}(n_i,{\mathrm{logit}}^{-1}{\theta }_i)$ and ${\theta }_i=x_i^T\beta +\lambda Z_i+{\eta }_i$, where $Z$ is a stationary unit-variance autoregressive process of order one with coefficient $a$ and the independent ${\eta }_i$ have normal distribution with variance $v$. The marginal latent variance is ${\lambda }^2+v$, and its off-diagonal covariance is ${\lambda }^2{a}^{|i-j|}$. Random success probabilities produce overdispersion relative to the binomial distribution.

For two independent binomial distributions with event and nonevent counts $a,b$ and $c,d$, the odds ratio is $ad/(bc)$. The delta method applied to $\log (p/(1-p))$, whose derivative is $1/[p(1-p)]$, gives a group variance $1/[Np(1-p)]$. Substitution of the fitted proportion gives $1/a+1/b$ in the first group and $1/c+1/d$ in the second. Their independence gives the displayed sum. A normal approximation then gives a confidence interval on the log odds ratio scale; exponentiation produces an odds-ratio interval. Zero cells require another approximation or model.

For a sample of size $n$ from a continuous probability distribution, this interval covers the median $F^{-1}(1/2)$ with probability $1-2{\sum }_{r=0}^{j-1}\left(\genfrac{}{}{0ex}{}{n}{r}\right)2^{-n}$, for $1\le j<n/2$. Count observations below the median: their number has the binomial distribution with parameters $n,1/2$. No strict increase of the distribution function is needed.

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.