The Triset Table as a Count-of-Events Calculator

The Triset Tabulation Theorem (Denewade, 2025) arranges the seven non-empty regions of a three-set Venn diagram in a table whose row totals are the set measures. This working paper identifies what that table computes, and finds that the answer is an object practitioners price every day. Let N be the number of the three events A, B, C that occur. The table's column totals are the distribution of N, and its grand total is E[N]. The Triset table is therefore the three-event case of the classical relation between binomial moments and the count distribution (Feller, 1968), equipped with a fill order that maps the seven quantities practitioners actually hold, three marginals, three pairwise joints and one triple, onto the eight elementary probabilities. What the table computes Writing X, Y, g and h for the exclusive, pairwise-only, common and outside regions, and S1, S2, S3 for the binomial moments: P(N = 1) = X · P(N = 2) = Y · P(N = 3) = g · P(N = 0) = hX + 2Y + 3g = S1 = E[N]P(N ≥ 1) = S1 − S2 + S3 · P(N ≥ 2) = S2 − 2S3 The first line is the table read column by column; the third is inclusion–exclusion together with its at-least-two analogue. The contribution is not the identity, which is old, but the layout: a table in which the count strata are columns, and a fill order that is hard to get wrong. The fill Algorithm 1 takes the seven joint-form inputs and returns all eight atoms by subtraction alone, with no simulation and no conditional probabilities: enter the row totals and the boxed pairwise sums, place the triple, subtract it to get the pairwise-only cells, mirror those into the first-copy column, and subtract each row to get the exclusive cells. Every subtraction must return a non-negative number, and a negative one identifies the offending input. The tree is downstream of the table, not an alternative to it A three-stage binary tree on A, B, C has eight leaves, and those leaves are the eight atoms. Its seven branch probabilities are as numerous as the inputs of Algorithm 1, but they are conditional and ordered. When data arrive in joint form, from a copula, a correlation matrix, a life table or a survey, the conditionals are not among the inputs: each is a quotient of table cells, for instance P(B | A) = (d + g) / p1 · P(C | A ∩ B) = g / (d + g) So the tree can only be drawn after the table has been filled. It is also committed to one ordering, where the table serves every ordering at once: any conditional P(E | F), for E and F Boolean combinations of the three events, is a ratio of two cell sums. Feasibility bounds for the unspecified triple In practice the pairwise joints are often known, from default correlations, bivariate copulas or two-way tabulations, while the triple is not. Because every cell is affine in g, requiring all eight to be non-negative gives the admissible range in closed form: gmin = max{0, p12+p13−p1, p12+p23−p2, p13+p23−p3}gmax = min{p12, p13, p23, 1 − S1 + S2} This is the n = 3 instance of Hailperin's (1965) best-possible bounds, which in general need a linear programme; here the programme has one free variable and reads straight off the table. The marginals and pairs are coherent exactly when the interval is non-empty, which makes the fill a cell-level coherence check: pairwise incoherence is caught at step 2, and three-way incoherence at step 6, where some exclusive cell is negative for every g. Inputs can be pairwise coherent and still admit no non-negative triple, the three-event face of the attainability problem for correlation matrices of dependent Bernoulli variables (Chaganty and Joe, 2006). A sharp corollary follows. Over the feasible interval, of width Δg, the first-, second- and third-to-default probabilities range over widths Δg, 2Δg and Δg. The second-to-default probability is twice as sensitive to the unspecified triple as the union probability, and more so in relative terms, because it is the smaller number. That is the quantitative version of a familiar trading-desk observation: second-to-default baskets are the hard ones to quote. Instruments that read off the cells nth-to-default basket credit default swaps. The trigger probability is P(N ≥ n), a column sum, and the eight atoms are the loss distribution of the basket, so three-name tranche expected losses are a lookup rather than a simulation. Multiple-life statuses on three lives. Joint-life is g, last-survivor is X + Y + g, at-least-two is Y + g, exactly-one is X. The seven-term last-survivor formula candidates learn to recite becomes the sum of a column block, and at-least-k-survivor annuities are discounted column sums. Clash reinsurance and multi-trigger catastrophe bonds. A clash cover, and a “two of three” parametric trigger, attach on P(N ≥ 2). Count-digital and worst-of equity notes. Full coupon on h, knock-in on 1 − h, reduced coupon on X, with marginals, pairs and triple supplied by Φ, Φ2 and Φ3 under Black–Scholes. Multi-morbidity rating and multi-line penetration. The column sums are the exactly-k-condition shares behind a multi-condition load, and the exactly-one / exactly-two / three-product customer shares behind a bundling discount, with S1 the average number of products per customer. Worked examples A three-name basket. Default probabilities 0.05, 0.08 and 0.10 with a flat default correlation of 0.20 and a triple of 0.006 give P(N ≥ 1) = 0.178, P(N ≥ 2) = 0.046 and P(N = 3) = 0.006, or 1,068, 276 and 36 basis points at a 60% loss given default. Had the triple been left unspecified, it would only be pinned to [0, 0.016]: enough to fix the first-to-default premium to within about 9% of itself, but not enough to fix the second-to-default premium to within a factor of two. A Gaussian copula calibrated to the same pairs puts the triple at 0.0078 and a Student-t copula with four degrees of freedom at 0.0077, so once the pairs are matched the copula family barely moves it. The exposure that matters is the decision to leave the triple open, or an error in a pair, which propagates into two cells. Three lives under dependent mortality. Survival probabilities 0.90, 0.85 and 0.80 with positively dependent pairs and a three-life probability of 0.66 give a joint-life status of 0.660, at-least-two of 0.900 and last-survivor of 0.990, against 0.612, 0.941 and 0.997 under independence with the same marginals. Dependence moves mass to the extremes, and by the corollary above the at-least-two status is the one most exposed to how the three-life probability was set. What the paper offers, and what it does not It offers a representation: a layout in which the count strata are columns, a fill order that is hard to get wrong, closed-form feasibility bounds with a cell-level coherence check, and a catalogue of instruments that read directly off the cells. It is explicit about its limits. The method covers three events: beyond a handful of names, the one-factor conditional-independence recursion (Andersen, Sidenius and Basu, 2003; Laurent and Gregory, 2005) is the method of choice. The atoms are unordered, recording which events occurred and not in what order, so contingent insurances and path-dependent payoffs need a time-ordered model. And the inputs must be joint-form: where they are conditional and sequential, a tree or a multiple-state model is the right tool and the table adds nothing. The identity is Feller's, the bounds are Hailperin's, and the instruments are standard. The contribution is representational, and the paper should be read as such. An appendix records corrigenda to the 2025 preprint, so the two papers can be read together. Author: Ali Denewade, Founder, Intelligent Actuaries.Keywords: inclusion–exclusion; binomial moments; contingency table; basket credit default swap; nth-to-default; multiple-life status; last survivor; feasibility bounds; coherence; Fréchet bounds.Indicative syllabi: SOA Exam P; FAM-L / ALTAM (multiple-life functions); QFI (credit derivatives, copulas).Status: Draft v0.1, circulated for comment.Project: intelligentactuaries.com/research/triset-count, where an interactive edition carries the Venn diagram, the tree and the sensitivity figure live in the browser.