A group of five applicants for a pair of identical jobs...

A group of five applicants for a pair of identical jobs consists of three men and two women. The employer is to select two of the five applicants for the jobs. Let $S$ denote the set of all possible outcomes for the employer's selection. Let $A$ denote the subset of outcomes corresponding to the selection of two men and $B$ the subset corresponding to the selection of at least one woman. List the outcomes in $A, \bar{B}, A \cup B, A \cap B,$ and $A \cap \bar{B}$. (Denote the different men and women by $M_{1}, M_{2}, M_{3}$ and $W_{1}, W_{2},$ respectively.)

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Key Concepts

Sample Space

The sample space is the complete set of all possible outcomes of a random experiment. In selection problems, it represents every way in which the selection (in this case, pairs of applicants) can be made from the total pool. Understanding the sample space is essential because it forms the environment in which all further event definitions, such as intersections and unions, are analyzed.

Combinatorial Counting

Combinatorial counting involves techniques like combinations and permutations to enumerate the number of ways selections can be made without having to list every outcome explicitly. In selection problems where order is irrelevant, combinations are especially useful in calculating how many subsets or groups can be chosen from a larger set. This provides the foundational tools to quantify elements in the sample space and in various events.

Set Operations

Set operations such as union, intersection, and complement are vital in analyzing relationships between different events in probability and combinatorics. The union of sets combines all outcomes that belong to at least one event, the intersection focuses on outcomes common to multiple events, and the complement includes all outcomes not contained in a particular event. Mastery of these operations allows one to articulate and solve problems involving overlapping criteria.

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