Math 1300: Section 8-1 Sample Spaces, Events, and Probability

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ExperimentsSample Spaces and Events

Probability of an EventEqually Likely Assumption

Math 1300 Finite MathematicsSection 8-1: Sample Spaces, Events, and Probability

Jason Aubrey

Department of MathematicsUniversity of Missouri

Jason Aubrey Math 1300 Finite Mathematics

Probability theory is a branch of mathematics that has beendeveloped do deal with outcomes of random experiments. Arandom experiment (or just experiment) is a situationinvolving chance or probability that leads to results calledoutcomes.

DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.

An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.We say that an event E occurs if any of the simple eventsin E occurs.

DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.

We say that an event E occurs if any of the simple eventsin E occurs.

DefinitionThe set S of all possible outcomes of an experiment a waythat in each trial of the experiment one and only one of theoutcomes (events) in the set will occur, we call the set S asample space for the experiment. Each element in S iscalled a simple outcome, or simple event.An event E is defined to be any subset of S (including theempty set and the sample space S). Event E is a simpleevent if it contains only one element and a compoundevent if it contains more than one element.We say that an event E occurs if any of the simple eventsin E occurs.

Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die.

Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).

(1,5) (2,2)

What is the sample space S for this experiment?

Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5)

or (Red=2, Green=2).

Example: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2).

(1,5) (2,2)

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

To clarify, the sample space is always a set of objects. In thiscase,

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

We can often use the counting techniques we learned in thelast chapter to determine the size of a sample space. In thiscase, by the multiplication principle:

n(S) = 6× 6 = 36

To clarify, the sample space is always a set of objects. In thiscase,

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

We can often use the counting techniques we learned in thelast chapter to determine the size of a sample space. In thiscase, by the multiplication principle:

n(S) = 6× 6 = 36

Events are subsets of the sample space:

A simple event is an event (subset) containing only oneoutcome. For example,

E = {(3,2)}

is a simple event.A compound event is an event (subset) containing morethan one outcome. For example,

E = {(3,2), (4,1), (5,2)}

is a compound event.

Events are subsets of the sample space:A simple event is an event (subset) containing only oneoutcome. For example,

E = {(3,2)}

is a simple event.

A compound event is an event (subset) containing morethan one outcome. For example,

E = {(3,2), (4,1), (5,2)}

Events are subsets of the sample space:A simple event is an event (subset) containing only oneoutcome. For example,

E = {(3,2)}

is a simple event.A compound event is an event (subset) containing morethan one outcome. For example,

E = {(3,2), (4,1), (5,2)}

Events will often be described in words, and the first step willbe to determine the correct subset of the sample space.

Forexample

“A sum of 11 turns up” corresponds to the event

E = {(5,6), (6,5)}.

Notice that n(E) = 2.“The numbers on the two dice are equal” corresponds tothe event

F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

Here we have n(F ) = 6.“A sum less than or equal to 3” corresponds to the event:

G = {(1,1), (1,2), (2,1)}

Here n(G) = 3

Events will often be described in words, and the first step willbe to determine the correct subset of the sample space. Forexample

E = {(5,6), (6,5)}.

Notice that n(E) = 2.

“The numbers on the two dice are equal” corresponds tothe event

F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

G = {(1,1), (1,2), (2,1)}

Here n(G) = 3

E = {(5,6), (6,5)}.

F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

Here we have n(F ) = 6.

“A sum less than or equal to 3” corresponds to the event:

G = {(1,1), (1,2), (2,1)}

Here n(G) = 3

E = {(5,6), (6,5)}.

F = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

G = {(1,1), (1,2), (2,1)}

Here n(G) = 3Jason Aubrey Math 1300 Finite Mathematics

Example: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.

(a) What is the sample space for this experiment?