Events

• Definition 2.2: An event is a subset of a sample space..
  • Null set, denoted $\varnothing$, contains no elements at all.
  • Example 2.4: Given the sample space $S= \{t \vert t\geq 0\}$, where $t$ is the life in years of a certain electronic component.
  • The event $A$ that the component fails before the end of the fifth year is the subset $A= \{t \vert 0 \leq t< 5\}$.
• Definition 2.3: The complement of an event $A$ with respect to $S$ is the subset of all elements of $S$ that are not in $A$. We denote the complement of $A$ by the symbol$A'$.
  • Example 2.5: Let $R$ be the event that a red card is selected from an ordinary deck of 52 playing cards.
  • $S$ be the entire deck.
  • $R'$ is the event that the card selected from the deck is not a red but a black card.

• Definition 2.4: The intersection of two events $A$ and $B$, denoted by the symbol $A \cap B$, is the event containing all elements that are common to $A$ and $B$.
• Example 2.7: Let $P$ be the event that a person selected at random while dining at a popular cafeteria is a taxpayer.
  • $Q$ is the event that the person is over 65 years of age.
  • The event $P \cap Q$ is the set of all taxpayers in the cafeteria who are over 65 years of age.
• Definition 2.5: Two events $A$ and $B$ are mutually exclusive, or disjoint if $A \cap B= \varnothing$, i.e., if $A$ and $B$ have no elements in common. Two events can not occur simultaneously.
• Example 2.9:
  • Let $A$ be the event that the program belongs to the NBC network.
  • Let $B$ be the event that the program belongs to the CBS network.
  • $A$ and $B$ are mutually exclusive.

• Definition 2.6: The union of the two events $A$ and $B$, denoted by the symbol $A \cup B$, is the event containing all the elements that belong to $A$ or $B$ or both.
• Example 2.12:
  • If $M= \{ x\vert 3< x< 9\}$ and $N= \{ y\vert 5< y< 12\}$, then
  • $M \cup N=\{ x\vert 3< x< 12\}$
• The relationship between events and the corresponding sample space can be illustrated graphically by Venn diagram.
• In a Venn diagram, let the sample space be a rectangle and represent events by circles. In Fig. 2.3

  Figure 2.3: Events represented by various regions.

  
  • $A \cap B$ : regions 1 and 2
  • $A \cup C$ : regions 1, 2, 3, 4, 5, and 7
  • $B' \cap A$ : regions 4 and 7