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Glacius
ProbabilityConcept reference

Partitions

A partition is a collection of nonempty pairwise-disjoint subsets whose union is the whole sample space.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A partition splits a sample space into nonempty parts
  3. You can check a partition from record destinations
  4. Check every pair of parts
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01A partition splits a sample space into nonempty parts#

A partition splits a sample space into nonempty parts. Every outcome belongs to exactly one part: there are no gaps and no overlaps.

For S = {a, b, c, d}, the parts {a, c}, {b}, {d} cover all four outcomes. None shares a member or is empty.

Within the full sample space S, part X contains a and c, Y contains b, and Z contains d. Every outcome appears exactly once and all three parts are nonempty. Enclosure areas are schematic.Within the full sample space S, part X contains a and c, Y contains b, and Z contains d. Every outcome appears exactly once and all three parts are nonempty. Enclosure areas are schematic.
Figure 1Within the full sample space S, part X contains a and c, Y contains b, and Z contains d. Every outcome appears exactly once and all three parts are nonempty. Enclosure areas are schematic.
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Check your reasoning

S={1, 2, 3}. Parts: {1, 2}; {2, 3}. Partition?

  1. ANo: overlapping parts.
  2. BYes: each outcome once.
  3. CNo: incomplete coverage.
Show answer and explanation
No: overlapping parts.

Shared outcomes: {2}.

02You can check a partition from record destinations#

You can check a partition from record destinations. Each record must enter one named bin, and every named bin must receive at least one record.

Check your reasoning

Bins: L, R. Destinations: a→L; b→R; c→R; d→L. Partition?

  1. ANo: overlap.
  2. BYes: each once.
  3. CNo: missing members.
Show answer and explanation
Yes: each once.

Each record once; both bins nonempty.

03Check every pair of parts#

Check every pair of parts. Three parts can overlap in pairs even if nothing lies in all three. Empty parts also fail the definition; unequal part sizes are allowed.

Check your reasoning

S={a, b}. Parts: {a}; {b}; {}. “Every outcome once, so yes.” Partition?

  1. AYes: each once.
  2. BNo: overlap.
  3. CNo: an empty part.
Show answer and explanation
No: an empty part.

Empty parts are not allowed.

Key takeaway

Check nonempty parts, no pairwise overlap and complete coverage. Every outcome must belong to exactly one part.

  • Check nonempty parts and exact single membership.

Sources & further reading

  1. [1]
    Pishro-Nik, Introduction to Probability: 1.2.2 Set Operations ↗Pishro-Nik, Introduction to Probability · Book

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Partitions.” Math behind ML. /learn/p-partition