Binomial Distribution

The binomial distribution is built from a single Bernoulli trial: one experiment with two outcomes, coded 1 for success with probability p and 0 for failure with probability 1 minus p. Stacking n of these trials, holding p constant and requiring independence, gives a random variable X equal to the total number of successes, which can take any integer value from zero to n.

The probability of exactly x successes is the number of distinct orderings that produce x successes, multiplied by the probability of any one such ordering. The ordering count is the combination of n items taken x at a time, written as n factorial divided by the product of x factorial and (n minus x) factorial. The probability of one specific sequence is p raised to the power x times (1 minus p) raised to the power (n minus x). Multiplying the two gives the binomial probability mass function. The combinatorial term is what candidates most often omit: three successes in five trials can occur in ten different orders, and all of them count.

Expected successes equal n times p, an intuitive result: each trial contributes p on average. Variance is n times p times (1 minus p), so standard deviation is the square root of that quantity. Variance is maximised at p equal to 0.5, where uncertainty about each trial is greatest, and collapses toward zero as p approaches either extreme, because outcomes become nearly deterministic.

Symmetry also depends on p. At p equal to 0.5 the distribution is symmetric about n divided by two. For p below 0.5 it is skewed right, since successes are bounded below by zero but the mass sits near the low end; for p above 0.5 it is skewed left. Increasing n while holding p fixed pushes the distribution toward a bell shape, which is the basis of the normal approximation. Note the distribution is discrete: probabilities attach to individual integers, and there is no meaningful probability of a non-integer number of successes.

Beyond option lattices, the binomial framework appears wherever repeated two-state events are counted. A portfolio of independent, identically rated loans produces a binomial number of defaults over a fixed horizon, which is the starting intuition behind credit loss distributions before correlation is introduced. Interest rate lattices used for callable bonds and embedded options share the same up-down architecture.

Performance evaluation uses it directly. If a manager has no skill, monthly outperformance versus a benchmark is a coin flip, so the count of winning months over a fixed window is binomial with p equal to 0.5; observing an unusually high count is evidence against the no-skill hypothesis. The same logic underlies tests of proportions, such as whether the share of firms beating earnings estimates differs from a historical base rate, and quality-control style checks on error rates in operational data.

Three tasks recur. First, compute an exact probability from the mass function, including the combination term, for small n. Second, compute cumulative and complementary probabilities: the probability of at most x successes is the sum of individual probabilities from zero through x, and the probability of at least x successes is most efficiently found as one minus the cumulative probability through x minus one. Third, state mean and standard deviation and use them to describe the distribution or to set up a normal approximation.

Equally important is diagnosing whether the model applies at all. Verify that the number of trials is fixed in advance, that each trial has exactly two outcomes, that p does not drift across trials, and that outcomes are independent. Correlated defaults, learning effects, changing market regimes and finite-population sampling all break at least one condition, and the correct response is a different distribution rather than an adjusted binomial.