Transcript Events
Chapter S6
Elementary probability
Learning Objectives
– Understand elementary probability concepts
– Calculate the probability of events
– Distinguish between mutually exclusive, dependent and
independent events
– Calculate conditional probabilities
– Understand and use the general addition law for probabilities
– Understand and apply Venn diagrams
– Understand and apply probability tree diagrams
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 1
1
Probability of events
Sample space
– When a statistical experiment is conducted, there
are a number of possible outcomes. These are
called sample space and are often denoted by S
A coin is tossed. The sample space is:
S = {head, tail}
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 2
2
Definitions for probability of events
An event is some subset of a
sample space.
A coin is tossed. Define an event A to be:
A = outcome is a head
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 3
3
Definitions for probability of events
The impossible event (or empty set)
is one that contains no outcomes.
It is often denoted by the Greek
letter (phi).
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 4
4
Impossible event
Example:
A hand of 5 cards is dealt from a deck. Let A
be the event that the hand contains 5 aces.
Since there are only 4 aces in the deck,
event A cannot occur. Hence A is an
impossible event.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 5
5
Definitions for probability of events
If A is an event, the probability that it occurs is
denoted by P(A).
The probability (or chance) that an event A
occurs is the proportion of possible outcomes
in the sample that yield the event A. That is:
P A
Number of outcomes that yield event A
Total number of possible outcomes
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 6
6
Definitions for probability of events
Two events A and B are said to
be mutually exclusive if they
cannot occur simultaneously
A = outcome is a head
B = outcome is a tail
Since A and B cannot both occur, the events are
mutually exclusive.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 7
7
Definitions for probability of events
Suppose that A1, A2, A3…An are n
mutually exclusive events then:
P(A1 or A2…or An) = P(A1) + P(A2) + …+ P(An)
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 8
8
Definitions for probability of events
Two events A and B are
independent if the occurrence of
one does not alter the likelihood
of the other event occurring.
Events that are not independent
are called dependent.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 9
9
Definitions for probability of events
Suppose that A1, A2, A3…An are n
independent events then:
P(A1) and P(A2)…and P(An) = P(A1) × P(A2) ×…P(An)
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 10
10
Definitions for probability of events
The complements of an event are
those outcomes of a sample space for
which the event does not occur.
Two events that are complements of
each other are said to be
complementary
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 11
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Definitions for probability of events
The probability that event A occurs, given
that an event B has occurred, is called the
conditional probability that A occurs given
that B occurs. The notation for this
conditional probability is P(AB).
For any two events, A and B, the following
relationship holds:
P A and B
P A B
PB
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 12
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General addition law
When two events are not mutually exclusive
we should use the following general
additional law:
P(A or B) = P(A) + P(B) - P(A and B)
Note: If the events A and B are mutually
exclusive, P(A and B) = 0.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 13
13
Venn diagrams
Sample spaces and events are often presented in a
visual display called a Venn diagram. While there are
several variations as to how these diagrams are
drawn, we will use the following conventions.
1. A sample space is represented by a rectangle.
2. Events are represented by regions within the
rectangle. This is usually done using circles (or parts
of circles).
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 14
14
Venn diagrams
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 15
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Venn diagrams
The union of two events A and B is the set of all
outcomes that are in event A or event B.
The notation is:
Union of event A and event B A B
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 16
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Venn diagrams
The intersection of two events A and B is the set of
all outcomes that are in both event A and event B.
The notation is:
Intersecti on of event A and event B A B
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 17
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Probability tree diagrams
A visual display of the probabilities using a
probability tree diagram.
Especially useful for determining probabilities
involving events that are not independent.
Conditional probabilities are the
probabilities on the second tier of branches.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 18
18
Probability tree diagrams
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 19
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