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5.1 Random Phenomena and Probability
With random phenomena, we can’t predict the individual
outcomes, but we can hope to understand characteristics
of their long-run behavior.
For any random phenomenon, each attempt, or trial,
generates an outcome.
Chapter 5
Randomness and Probability
We use the more general term event to refer to outcomes
or combinations of outcomes.
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5.1 Random Phenomena and Probability
5.1 Random Phenomena and Probability
Sample space is a special event that is the collection of all
possible outcomes.
We denote the sample space S or sometimes Ω.
The Law of Large Numbers (LLN) states that if the events
are independent, then as the number of trials increases,
the long-run relative frequency of an event gets closer and
closer to a single value.
The probability of an event is its long-run relative
frequency.
Empirical probability is based on repeatedly observing the
event’s outcome.
Independence means that the outcome of one trial doesn’t
influence or change the outcome of another.
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5.2 The Nonexistent Law of Averages
5.3 Different Types of Probability
Many people confuse the Law of Large numbers with the
so-called Law of Averages that would say that things have
to even out in the short run.
Model-Based (Theoretical) Probability
The (theoretical) probability of event A can be computed
with the following equation:
The Law of Averages doesn’t exist.
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5.3 Different Types of Probability
5.4 Probability Rules
Personal Probability
Rule 1
A subjective, or personal probability expresses your
uncertainty about the outcome.
If the probability of an event occurring is 0, the event can’t
occur.
Although personal probabilities may be based on
experience, they are not based either on long-run relative
frequencies or on equally likely events.
If the probability is 1, the event always occurs.
For any event A,
.
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5.4 Probability Rules
5.4 Probability Rules
Rule 2: The Probability Assignment Rule
Rule 3: The Complement Rule
The probability of the set of all possible outcomes must
be 1.
The probability of an event occurring is 1 minus the
probability that it doesn’t occur.
where S represents the set of all possible outcomes and is
called the sample space.
where the set of outcomes that are not in event A is called
the “complement” of A, and is denoted AC.
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5.4 Probability Rules
5.4 Probability Rules
Rule 4: The Multiplication Rule
Rule 5: The Addition Rule
For two independent events A and B, the probability that
both A and B occur is the product of the probabilities of the
two events.
Two events are disjoint (or mutually exclusive) if
they have no outcomes in common.
The Addition Rule allows us to add the probabilities of
disjoint events to get the probability that either event
occurs.
provided that A and B are independent.
where A and B are disjoint.
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5.4 Probability Rules
5.5 Joint Probability and Contingency Tables
Rule 6: The General Addition Rule
The General Addition Rule calculates the probability that
either of two events occurs. It does not require that the
events be disjoint.
Events may be placed in a contingency table such as the
one in the example below.
Example: As part of a Pick Your Prize Promotion, a store
invited customers to choose which of three prizes they’d
like to win. The responses could be placed in the following
contingency table:
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5.5 Joint Probability and Contingency Tables
Marginal probability depends only on totals found in
the margins of the table.
5.5 Joint Probability and Contingency Tables
In the table below, the probability that a respondent chosen
at random is a woman is a marginal probability.
P(woman) = 251/478 = 0.525.
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5.5 Joint Probability and Contingency Tables
Joint probabilities give the probability of two events
occurring together.
5.5 Joint Probability and Contingency Tables
Each row or column shows a conditional distribution given
one event.
In the table above, the probability that a selected customer
wants a bike given that we have selected a woman is:
P(bike|woman) = 30/251 = 0.120.
P(woman and camera) = 91/478 = 0.190.
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5.6 Conditional Probability
5.6 Conditional Probability
In general, when we want the probability of an event from
a conditional distribution, we write P(B|A) and pronounce it
“the probability of B given A.”
A probability that takes into account a given condition is
called a conditional probability.
Rule 7: The General Multiplication Rule
The General Multiplication Rule calculates the probability
that both of two events occurs. It does not require that the
events be independent.
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5.6 Conditional Probability
5.7 Constructing Contingency Tables
Events A and B are independent whenever P(B|A) = P(B).
If you’re given probabilities without a contingency table,
you can often construct a simple table to correspond to the
probabilities and use this table to find other probabilities.
Independent vs. Disjoint
For all practical purposes, disjoint events cannot be
independent.
Don’t make the mistake of treating disjoint events as if they
were independent and applying the Multiplication Rule for
independent events.
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5.7 Constructing Contingency Tables
5.7 Constructing Contingency Tables
Example: A survey classified homes into two price
categories (Low and High). It also noted whether the houses
had at least 2 bathrooms or not (True or False). 56% of the
houses had at least 2 bathrooms, 62% of the houses were
Low priced, and 22% of the houses were both. Translating
the percentages to probabilities, we have:
The 0.56 and 0.62 are marginal probabilities, so they go in
the margins.
The 22% of houses that were both Low priced and had at
least 2 bathrooms is a joint probability, so it belongs in the
interior of the table.
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5.7 Constructing Contingency Tables
What Can Go Wrong?
•  Beware of probabilities that don’t add up to 1.
•  Don’t add probabilities of events if they’re not disjoint.
•  Don’t multiply probabilities of events if they’re not
independent.
Because the cells of the table show disjoint events, the
probabilities always add to the marginal totals going across
rows or down columns.
•  Don’t confuse disjoint and independent.
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What Have We Learned?
What Have We Learned?
•  Some basic rules for combining probabilities of outcomes
to find probabilities of more complex events:
•  Probability is based on long-run relative frequencies.
1) Probability for any event is between 0 and 1
•  The Law of Large Numbers speaks only of long-run
behavior and should not be misinterpreted as a law of
averages.
2) Probability of the sample space, S, the set of possible
outcomes = 1
3) Complement Rule
4) Multiplication Rule for independent events
5) General Addition Rule
6) General Multiplication Rule
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