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 Incoming links: Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik / 6.2.1 The Union Bound and Extension

Boole's inequality 2026-10-14

Boole's inequality, also called the union bound, bounds the probability that at least one of several events occurs by adding their individual probabilities. For a finite or countable family of events

A_{i}

in the same probability space,

P (⋃_{i} A_{i}) \leq min {1, \sum_{i} P (A_{i})} .

The union includes outcomes where several events occur together. There is no independence assumption: adding the individual probabilities counts such outcomes repeatedly, which can only make the sum larger. The cap at one uses the total mass of a probability measure. For infinitely many events, the sum is a nonnegative series, possibly infinite; an infinite sum gives only the bound one. This is countable subadditivity of a measure specialized to a probability measure. See Probability: Theory and Examples (fifth edition) by Rick Durrett (Section 1.1, Theorem 1.1.1(ii), printed p. 2).

For example, choose uniformly from the twelve outcomes

Ω = {1, \dots, 12}

, using the discrete uniform distribution. Put

A = {1, 2, 3, 4}

. If

B = {5, 6, 7, 8}

, these are disjoint events and their union has probability

8/12 = 2/3

, exactly the union bound. If

B = {3, 4, 5, 6}

, the union has only six outcomes, so its exact probability is

1/2

, although both individual probabilities are still

1/3

and the bound is still

2/3

. If

B = A

, the exact probability falls to

1/3

with the same bound. These examples isolate overlap in the union bound by keeping the individual probabilities fixed.

Figure 1.
The same individual event probabilities give different exact union probabilities as overlap increases
. Each panel is a fully specified discrete uniform distribution on twelve numbered outcomes. Blue means $A$ only, orange means $B$ only, purple means their intersection, and white means neither. The bottom bars compare the exact union probability with the sum of the individual probabilities. The three unions contain eight, six, and four outcomes; the sum counts eight in every panel. Details.

The union bound is useful when each event is easy to control but their joint behavior is difficult to determine. A short proof uses indicator functions: at every outcome,

1_{⋃_{i = 1}^{n} A_{i}} \leq \sum_{i = 1}^{n} 1_{A_{i}} .

Taking expected values and using linearity of expectation proves the finite bound.^[ref] For a countable family, the finite unions increase to the full union; continuity of probability passes the finite inequalities to the limit. The closure of a sigma-algebra under countable unions ensures that this limiting union is an event. For the continuity of probability step, see Probability: Theory and Examples (fifth edition) by Rick Durrett (Section 1.1, Theorem 1.1.1(iii), printed p. 2).

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