boole's inequality: meaning, definition, pronunciation and examples
C2technical/academic
Quick answer
What does “boole's inequality” mean?
A theorem in probability theory stating that for any finite or countable set of events, the probability that at least one event occurs is at most the sum of their individual probabilities.
Audio
Pronunciation
Definition
Meaning and Definition
A theorem in probability theory stating that for any finite or countable set of events, the probability that at least one event occurs is at most the sum of their individual probabilities.
A fundamental inequality used in probability, statistics, and measure theory, providing an upper bound for the probability of a union of events. It is also known as the union bound and is foundational for proofs in areas like algorithm analysis and information theory.
Dialectal Variation
British vs American Usage
Differences
No significant differences in usage, spelling, or meaning. The apostrophe-s ('s) is universally retained.
Connotations
Purely technical and neutral in both varieties.
Frequency
Equally low-frequency in both, confined to advanced mathematics, statistics, computer science, and engineering publications.
Grammar
How to Use “boole's inequality” in a Sentence
Boole's inequality states that P(∪ A_i) ≤ Σ P(A_i).By Boole's inequality, we can bound the error probability.A direct application of Boole's inequality yields the result.Vocabulary
Collocations
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in graduate-level probability, statistics, theoretical computer science, and electrical engineering courses and papers.
Everyday
Never used in everyday conversation.
Technical
Essential in probabilistic analysis, algorithm design, communication theory, and mathematical proofs involving unions of events.
Vocabulary
Synonyms of “boole's inequality”
Neutral
Weak
Watch out
Common Mistakes When Using “boole's inequality”
- Misspelling as 'Boolean inequality' (which refers to inequalities in Boolean algebra, not the probability theorem).
- Confusing it with the Bonferroni inequalities (which are a more precise set of bounds).
- Incorrectly applying it to the intersection of events instead of the union.
FAQ
Frequently Asked Questions
No, a key strength of Boole's inequality is that it holds for any events, whether they are independent or not.
It is named after George Boole (1815–1864), the English mathematician who laid the foundations of Boolean algebra, although the inequality itself is a basic result in measure theory.
Yes, the countable version of Boole's inequality applies to a countably infinite set of events.
Boole's inequality is the first step (a simple upper bound) in the more precise inclusion-exclusion principle, which alternately adds and subtracts intersection probabilities.
A theorem in probability theory stating that for any finite or countable set of events, the probability that at least one event occurs is at most the sum of their individual probabilities.
Boole's inequality is usually technical/academic in register.
Boole's inequality: in British English it is pronounced /buːlz ˌɪnɪˈkwɒlɪti/, and in American English it is pronounced /buːlz ˌɪnɪˈkwɑːləti/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of George Boole (Boolean logic) giving you a simple upper 'BOOLE' (like a 'boo' limit) for combining probabilities: just add them up, but know it's a generous (not tight) bound.
Conceptual Metaphor
SAFETY NET: It provides a guaranteed upper limit (safety net) for the combined chance of several things happening, even if you don't know how they interact.
Practice
Quiz
What is the primary use of Boole's inequality?