Download Chapter 6 Elementary Probability

What is the Probability of an Event?
Probability is the study of how likely it is that an event will occur.
Chapter 6
Elementary Probability
Probability terms include:
• Trial – A single act by which an observation is noted, e.g. roll of a die.
• Experiment - The process by which an observation is noted and consists of
one or more trials.
• Event – A subset of all the possible outcomes of an experiment E.g. Rolling a 6.
• Sample space – The set of all feasible outcomes of an experiment. E.g. when
rolling a die the sample space is: (1, 2, 3, 4, 5, 6).
• Define the Probability of an event
• Distinguish between the three Types of Probability:
• Classical probability
• Relative frequency probability
• Subjective probability
• Explain Set Operations and Venn Diagrams
• Distinguish between Mutually Exclusive, Independent and Dependent Events
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What is the Probability of an Event?
What is the Probability of an Event?
For a set of all possible outcomes (S) with each outcome equally likely to occur,
the probability of an event (E) occurring is equal to the number of ways that
event can occur divided by the total number of possible outcomes.
If an event has 0 probability it can never occur. If it has a probability of 1 it will
always occur.
The sum of the probabilities of each event in the sample space occurring will
always equal 1.
Probabilities are usually expressed in decimals or percentages. E.g. if an event
has a probability of 0.52 then it has a 52% chance of occurring.
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Classical Probability – Straightforward mathematical approach assigns the
chance that an event will occur to the number of possible results.
Relative Frequency Probability – Examines previous data to assist with current
decisions.
Subjective Probability – It refers to decision making where perhaps less direct
data exists. Final decisions are made on a subjective basis. i.e. When will the
next election be held?
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Set Operations and Introduction to Venn
Diagrams
Three Types of Probability
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• A set is a collection of objects or elements. Elements are shown inside the
parenthesis { }.
• The set of all elements is called the sample space, usually denoted ‘S’.
• The elements of the set are all the possible outcomes of an experiment.
• Once an outcome has occurred it is called an event.
• A subset of the sample space will contain some of the elements of S. A
subset can also be called an event.
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Set Operations and Introduction to Venn
Diagrams
Venn Diagrams
Sets can be displayed graphically through the use of Venn Diagrams.
• A rectangle is drawn to represent the sample space and a point is assigned
to represent each simple event.
• Circles are drawn around the sample points to represent events.
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Set Operations - Union
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Set Operations - Union
Union – The union of two events is the event containing all elements in the
two sets.
e.g. If X and Y are two events in the sample space S, the union of X and Y is
the event containing all sample points in X and Y. The Union is written as ∪.
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Set Operations - Complement
Mutually Exclusive Events
The complement of a set contains all the elements is a sample which are not in
the set itself.
Events are mutually exclusive if the occurrence of any one of the events
excludes the occurrence of others, e.g. flipping a head excludes the flipping of
a tail.
In other words the complement of an event happening is the event not
happening.
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Mutually exclusive events have no overlapping area in the Venn diagram.
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Independent Events
Dependent Events
Two or more events are said to be independent if the occurrence or non
occurrence of one of them in no way effects the occurrence or non-occurrence
of the others.
Two or more events are dependent when the probability of one event taking
place is subject to another event taking place, e.g. the probability of rain and
a cloudy sky. It is more likely to rain when the sky is cloudy.
The events are unconnected, e.g. two consecutive coin tosses, the result of the
first cannot impact the result of the second.
When two events A and B are dependent, the probability of event A occurring
given that B has already occurred is the probability of events A and B
occurring together divided by the probability of event B.
When two events are independent the probability that they will both occur is
given by multiplying the probabilities that each will occur together:
P(A and B) = P(A) x P(B)
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