Improved Bounds on the Probability of a Union and on the Number of Events that Occur
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| Format: | Preprint |
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2025
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| _version_ | 1866909615020572672 |
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| author | Adler, Ilan Karp, Richard M. Ross, Sheldon M. |
| author_facet | Adler, Ilan Karp, Richard M. Ross, Sheldon M. |
| contents | Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question dates back to Boole in the 19th century, and it is well known that the odd partial sums of the Inclusion- Exclusion formula provide upper bounds, while the even partial sums provide lower bounds. We give a combinatorial characterization of the error in these bounds and use it to derive a very simple proof of the strongest possible bounds of a certain form, as well as a couple of improved bounds. The new bounds use more information than the classical Bonferroni-type inequalities, and are often sharper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds on the Probability of a Union and on the Number of Events that Occur Adler, Ilan Karp, Richard M. Ross, Sheldon M. Combinatorics Statistics Theory Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question dates back to Boole in the 19th century, and it is well known that the odd partial sums of the Inclusion- Exclusion formula provide upper bounds, while the even partial sums provide lower bounds. We give a combinatorial characterization of the error in these bounds and use it to derive a very simple proof of the strongest possible bounds of a certain form, as well as a couple of improved bounds. The new bounds use more information than the classical Bonferroni-type inequalities, and are often sharper. |
| title | Improved Bounds on the Probability of a Union and on the Number of Events that Occur |
| topic | Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2505.12243 |