Set Operations with Events
Since events are subsets of a sample space, they are intrinsically sets of outcomes or sample points. Thus, events obey the rules of set operations. Events can be depicted as subsets of their sample spaces using Venn diagrams. Given that event A and event B are subsets of the sample space S, the following definitions apply:
The union of events A and B, denoted by A È B , is the set of all sample points in S that belong to event A, event B, or both.
The intersection of events A and B, denoted by A Ç B , is the set of all sample points in S that belong to both event A and event B.
Mutually exclusive events are events whose intersection is the null space, i.e., if event A and event B are mutually exclusive, then A Ç B = Æ . Mutually exclusive events have no outcomes or sample points that are in common with each other.
The complement of event A (with respect to S), denoted by A' , is the set of all sample points in S that do not belong to event A. The event A and its complement A' are mutually exclusive events, since their intersection is an empty set.
A partition of the sample space S is a set of events A1 , A2 ,....., An such that the following two conditions are true:
(1) AiÇ Aj = Æ for i, j Î{1, 2, ...., n }, i ¹ j
(2) (A1È A2È ......... È An ) = S
The event subsets of the sample space S that form a partition of S are mutually exclusive.
EX. Each team in a basketball league plays 20 games in one tournament. Event A is the event that Team 1 wins 15 or more games in the tournament. Event B is the event that Team 1 wins less than 10 games. Event C is the event that Team 1 wins between 8 to 16 games. Of course, Team 1 can win at most 20 games.
A È B is the event that Team 1 wins 15 or more games or wins 9 or less games. A Ç B is the null space, since Team 1 cannot win 15 or more games and have less than 10 wins at the same time. Therefore, event A and event B are mutually exclusive events.
A È C is the event that Team 1 wins at least 8 games. A Ç C
is the event that Team 1 wins 15 or 16 games.B È C is the event that Team 1 wins at most 16 games.
B Ç C is the event that Team 1 wins 8 or 9 games.A' is the event that Team 1 wins 14 or fewer games.
B' is the event that Team 1 wins 10 or more games.
C' is the event that Team 1 wins fewer than 8 or more than 16 games.
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