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Basic Probability

What’s in a problem?

What’s in a research question?

Hypotheses

Operationalizing Concepts
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
Are the things that we observe different from
what would be expected by chance?
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
Myers and Chen (1996) conducted a
study where they tested teenagers
who had previously participated in an
experiment 12 years earlier. The
teenagers were presented with
objects, only one of which they had
seen 12 years ago. Suppose we
present them with one object they had
seen and one they had not seen.

How do we determine if their choice
responses are ‘real’ or just chance?
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
Probability Distribution
 What would the distribution be
like if it were only due to chance?

Decision Rule
 What criteria do we need in order
to determine whether an
observation is just due to chance
or not.
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
The probability of an
outcome is the relative
frequency that an
event can be expected
to occur.

The probability
distribution is the set
of relative frequencies
for every possible
outcome.
Probability distribution of
number of correct responses for
4 teenagers when p(correct) = .5
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
Basic Concepts in Probability

Basic Probability Rules

Special Types of Probability
 Joint Probabilities
 Probabilities of Unions of Events
 Conditional Probabilities
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
Random Selection
 Every possibility has equal chance of being
chosen.

Independence
 The probability of a response on one trial
does not depend on the outcome of any
other trials.

Elementary Event
 Possible outcomes of a probability
experiment
 E.g., each coin toss

Sample Space
 The complete set of elementary events
 E.g., all coin tosses
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
Mutually exclusive events
 Two or more events that
cannot occur at the same time.

Exhaustive events
 A set of events that accounts
for all of the elementary events
in the sample space.
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
Multiplication Rule
 For independent events, we can multiply the
probabilities together to get the probability for all
of the events occurring.
▪ Example: Probability of rolling a die twice and getting 6
on both rolls.
▪ But what happens if the events are not independent?
▪ Example: probability of selecting a club from a deck of cards, then
selecting another club (without replacement)?
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
when two or more events will happen at the same time, and
the events are independent, then the special rule of
multiplication law is used to find the joint probability:
P(X and Y) = P(X) x P(Y)

when two or more events will happen at the same time, and
the events are dependent, then the general rule of
multiplication law is used to find the joint probability:
P(X and Y) = P(X) x P(Y|X)
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
The addition rule
 For independent events, we can add the
probabilities to get the probability of either event
occurring.
▪ Example: Rolling die and getting a 4 or a 6.
▪ Again, what happens if the events are not independent
(in this case, mutually exclusive)?
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
When two or more events will happen at the same time, and the events
are mutually exclusive, then:
P(X or Y) = P(X) + P(Y)

When two or more events will happen at the same time, and the events
are not mutually exclusive, then:
P(X or Y) = P(X) + P(Y) - P(X and Y)
For example, what is the probability that a card chosen at random from a
deck of cards will either be a king or a heart?

P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77%
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
Joint Probabilities

Probabilities of Unions of Events

Conditional Probabilities
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
Probability of obtaining a particular combination of
events.
 E.g., probability of flipping a coin twice and getting heads
both times. Just use multiplication rule!
▪ P (A and B) = n(A and B) / n (S)
 What about non-independent events?
▪ E.g., probability of a given respondent in a survey being female and
voting for reproductive rights?
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
A union of two elementary events consists of all the
elementary events belonging to either of them.

Examples:
 Probability of flipping a coin and it being heads or tails.
(mutually exclusive union)
 Non-independent events: (pres_data.dta) Probability of
being a female or rating the president with a 4.
▪ p(E1) + p(E2) – p(E1 and E2)
▪ (.48) + (.16) – (.06) = .58
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
Probability of an event occurring given that another
event has occurred.
 Example: probability of rating the president as 4 (agree)
given that the respondent is a female.
▪ Example: pres_data.dta
 P(E1|E2) = 6/48 = .125

3 Doors Problem
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
Any given sample may not be representative
of the population, even if randomly drawn.

Non-representative samples will lead to
erroneous statements about the population

However…basic probability theory can helps
us determine what the population probably
looks like based on our sample.
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
Obtaining a representative sample
 1930’s Literary Digest poll:
▪ Franklin Roosevelt predicted to lose the 1936
presidential election by a landslide.
▪ Oops… he won by a landslide.

Accounting for outliers in sample
 What is the best undergraduate major if you
want a high income (UNC-Chapel Hill survey)?
▪ One outlier, a geography major named
Michael Jordan, accounted for the huge skew
in average salaries for graduates (at the time
he made $80 million/year)
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