Partition of a Set¶

Definition¶

A partition of a set $E$ is a family $\{B_i\colon i\in I\}$ of nonempty subsets satisfying

B_i\cap B_j=\varnothing\quad(i\ne j),\qquad\bigcup_{i\in I}B_i=E.

In probability, take $E=S$ $S$ , the sample space, and require the blocks to be events. The elementary total-probability sums use finite or countable partitions. A nonempty event can still have probability zero; omit such blocks from formulas that divide by their probability.

Interpretation¶

A partition splits a set into nonempty blocks that do not overlap and together cover everything, so every element lands in exactly one block. In probability, a partition of the sample space is a complete list of cases without overlaps: every outcome belongs to exactly one case.

A rectangle for the sample space divided into four vertical strips B1 to B4, which do not overlap and together cover it.

The four strips $B_1,\ldots,B_4$ satisfy $B_i\cap B_j=\varnothing$ for $i\neq j$ and $B_1\cup\cdots\cup B_4=S$ $S$ , so additivity gives $\mathbb{P}(B_1)+\cdots+\mathbb{P}(B_4)=1$ $1$ . Any event $A$ is cut by the strips into the pieces $A\cap B_i$ $B_{i}$ , which are again disjoint with union $A$ , so $\displaystyle \mathbb{P}(A)=\sum_i\mathbb{P}(A\cap B_i)$ $B_{i})$ : a question is split into cases.

Both conditions are needed for this sum. If two blocks overlapped, outcomes in the overlap would be counted twice; if the blocks did not cover the whole sample space, outcomes outside all blocks would be missed, so the sum could be too small. The family $\{B,B^c\}$ is a partition when the event $B$ and its complement are both nonempty; otherwise the empty block is omitted, leaving $\{S\}$ . Other examples are the six values of a die or the possible numbers of defective items in a batch.

Writing $\mathbb{P}(A\cap B_i)=\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)$ for positive-probability blocks turns the sum into the law of total probability, omitting zero-probability contributions. It in turn supplies the denominator of Bayes' rule. In Math II, the equivalence classes of an equivalence relation form a partition of the set, and every partition arises in this way.

Key Results¶

Every outcome belongs to exactly one partition event.
For any event $A$ , the events $A\cap B_i$ are disjoint and have union $A$ .

Examples¶

For a fair die, the odd and even outcomes partition $\{1,\ldots,6\}$ .
For a count $N$ taking finite nonnegative integer values, the nonempty events among $\{N=0\},\{N=1\},\ldots$ partition the sample space. Nonempty blocks may still have probability zero.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 2
Nicky van Foreest · Probability Distributions · Week 3

Aliases: set partition Keywords: partition of the sample space, case split, Partition Symbols: element of, intersection, sum, union

Last updated: 27 October 2026 11:52