Suppose a family contains two children of different ages,...

Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let $F$ denote that a child is female and $M$ that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set $S$ of possible observations: $$S=\{F F, F M, M F, M M\}$$ Let $A$ denote the subset of possibilities containing no males; $B$, the subset containing two males; and $C,$ the subset containing at least one male. List the elements of $A, B, C, A \cap B, A \cup B, A \cap C$ $A \cup C, B \cap C, B \cup C,$ and $C \cap \bar{B}$

Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let $F$ denote that a child is female and $M$ that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set $S$ of possible observations:
$$S=\{F F, F M, M F, M M\}$$
Let $A$ denote the subset of possibilities containing no males; $B$, the subset containing two males; and $C,$ the subset containing at least one male. List the elements of $A, B, C, A \cap B, A \cup B, A \cap C$
$A \cup C, B \cap C, B \cup C,$ and $C \cap \bar{B}$

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Key Concepts

Sample Space

The sample space is the complete set of all possible outcomes in a given experiment or scenario. It provides the foundational context for probability problems by enumerating every possible result, which then allows for systematic analysis of events and probabilities.

Events

An event is defined as a subset of the sample space and represents a collection of outcomes that share a common property or satisfy a given condition. Defining events helps in structuring probability problems, where the focus is on determining how likely it is for these events to occur.

Set Operations

Set operations, including union, intersection, and complement, are fundamental concepts in probability and set theory. The union of sets combines all outcomes from multiple events, the intersection identifies outcomes common to two or more events, and the complement includes all outcomes not found in a specified event. These operations are essential for manipulating and understanding relationships between different events.

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