binomial distribution
The binomial distribution counts how many successes you get when you repeat the same yes-or-no trial a fixed number of times. Flip a fair coin 10 times: how many heads? Insure 1,000 identical lives: how many die this year? Whenever you have a fixed number of independent trials, each with the same success probability, the count of successes follows a binomial distribution.
It has two parameters: n, the number of trials, and p, the success probability on each. The probability of getting exactly k successes is C(n,k) · p^k · (1-p)^(n-k), where C(n,k) counts the ways to choose which k of the n trials succeed. Its mean is np (on average, that fraction of trials succeed) and its variance is np(1-p). A binomial is simply the sum of n independent Bernoulli trials, which is why its mean and variance are just n times the Bernoulli's.
The binomial is the natural model for a closed group of similar, independent risks over one period: deaths in a block of policies, defaults in a loan pool, claims among a fixed set of insureds. Actuaries use it to find the probability the number of claims exceeds some level, which feeds into capital and reinsurance decisions. Its key assumptions — a fixed n, constant p, and independence — are also its limits: when p varies between lives or the number at risk is itself uncertain, you reach instead for the Poisson or negative binomial.
Insure 1,000 independent lives each with death probability 0.02. The number of deaths is Binomial(1000, 0.02): expected 20 deaths, variance 1000 × 0.02 × 0.98 ≈ 19.6, standard deviation about 4.4. So 'around 20, give or take 4 or 5' is the planning range.
A block of identical, independent lives produces a binomial number of deaths.
The binomial assumes every trial shares the same p and all are independent; if the lives differ in risk or deaths cluster (an epidemic), the real spread is wider than np(1-p) suggests.