If the question of interest is
- "What is the suit of the chosen card?" S={heart, club, spade, diamond}
- "Is suit of the chosen card a heart?" S={yes, no}
- "How many hearts were chosen?" S={0,1}
The Language of Probability
Probability theory describes the likelihood of different outcomes or distinct results from chance processes (sometimes referred to as experiments.) The set of all the possible outcomes of an experiment is called the sample space.
The sample space, S, is the set of all possible outcomes of a chance process.
Example: The sample space for the outcome of a roll of a standard die is $S={1,2,3,4,5,6}$.
There may be more than one possible sample space for a given experiment. The way in which the sample space is described depends on the question of interest.
Events
Suppose that, when rolling a die, we are interested in a collection of outcomes rather than a single outcome. For instance, rather than being interested in whether a 2 was rolled, we might be interested in whether an even number was rolled.→ An event, A, is a collection of outcomes (or subset) of the sample space S . We write $A\subseteq S$.
NOTATION: Events are usually denoted by capital letters from the beginning of the alphabet: A, B, C...
Example: When a die is rolled, the sample space for the outcome is $S=\{1,2,3,4,5,6\}.$
Some events in S are:
- $A = \{2, 4, 6\}$
- $B= \{1, 2\}$
- $S=\{1,2,3,4,5,6\}$
- $\emptyset=\{\}$
Let B be the event the student gets at least one answer correct. (There are many other possibilities.)
Combinations of Events
We are often interested in combinations of events such as complements, unions, and intersections.
These combinations are themselves events.
A Venn diagram, like that shown, is helpful for seeing relationships between events.
→ A Venn Diagram uses a box and circles to illustrate relationships among events.
Union
The union of events A and B, A ∪ B, consists of all outcomes in S that are in A or B. The 'or' in this definition is a 'non-exclusive or' meaning that an outcome that is in both A and B is also in the union.
The shaded region in the Venn diagram represents the union of events A and B.
→ The union of events A and B consists of all outcomes in either A or B (or both).
Example: A card is randomly drawn from a deck of playing cards. Let C be the event the chosen card
is a club and let D be the event that the card is an ace.
C ∪ D is the event that the card is an ace or a club or both.
C ∪ D is the event that the card is an ace or a club or both.
If each outcome is represented as a two digit number where the first digit is the result of the first die and the second digit is the result of the second die. Then A ∪ B = {11,12,13,14,15,16,21,22,23,24,25,26,31,41,51,61,32,42,52,62}.
Intersection
The intersection of events A and B, A ∩ B, consists of all outcomes that are in S that are in both A and B.
The shaded region in the Venn diagram represents the intersection of events A and B.
→ The intersection of events A and B consists of all outcomes in both A and B.
Example: A card is randomly drawn from a deck of playing cards. Let C be the event the chosen card
is a club and let D be the event that the card is an ace. C ∩D is the event that the card is an ace and a club.
If each outcome is represented as a two digit number where the first digit is the result of the first die and the second digit is the result of the second die. Then A ∩ B = {12,21}.
Mutually Exclusive
Events A and B are mutually exclusive if there is no outcome that is in both A and B. If A and B are mutually exclusive and the outcome of an experiment is in event A then it is not in event B.
The Venn diagram shows mutually exclusive events A and B.
→ Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common, that is A ∩ B = ∅.
NOTATION: ∅ denotes the empty set. That is ∅ = {}.
Example: A card is randomly drawn from a deck of playing cards. Let C be the event the chosen card
is a club and let D be the event that the card is a heart. C and D are mutually exclusive.
If the card chosen is a club (event C) then it cannot be a heart (event D). Thus event C excludes event D and vice versa. Therefore C and D are mutually exclusive.
If the card chosen is a club (event C) then it cannot be a heart (event D). Thus event C excludes event D and vice versa. Therefore C and D are mutually exclusive.
There are a number of possibilties. Here are a couple:
- The event that a 3 is rolled is mutually exclusive with A.
- The event that the result is odd is mutually exclusive with A.
Complement
The complement of events A, $A'$ (sometimes Ac), consists of all outcomes that are in S that are in not in A. $A\cap A'=\emptyset$, that is, A and its complement are mutually exclusive. $A \cup A' = S$, that is, together A and its complement comprise the entires sample space.
The shaded region in the Venn diagram is the complement of event A.
→ The complement of event A consists of all outcomes in S that are not in A.
Example: A card is randomly drawn from a deck of playing cards.
Let C be the event the chosen card is a club.
C' is the event that the chosen card is not a club or C' is the event that the chosen card is a heart, diamond, or spade.
C' is the event that the chosen card is not a club or C' is the event that the chosen card is a heart, diamond, or spade.
A': At least one die result is not a one.