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Chapter 5

# PSYB07H3 Chapter 5: Chapter #5 - Probability (Lec 4)

Department
Psychology
Course Code
PSYB07H3
Professor
Dwayne Pare
Chapter
5

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Chapter #5 Lec 4
Chapter #5 Basic Concepts of
Probability
5.1 – Probability
- The oldest and most common definition of probability is called the analytic view
-Analytic view: Definition of probability in terms of analysis of possible outcomes
oIf an event can occur in A ways and can fail to occur in B ways, and if all possible
ways are equally likely (e.g., each M&M in the bag has an equal chance of being
drawn), then the probability of its occurrence is A/(A 1 B), and the probability of
its failing to occur is B/(A 1 B).
oBecause there are 24 ways of drawing a blue M&M (one for each of the 24 blue
M&M’s in a bag of 100 M&M’s) and 76 ways of drawing a different color, A 5 24,
B 5 76, and p(A) 5 24/(24 1 76) 5 .24.
-Frequentist view
oSample with replacement: Sampling in which the item drawn on trial N is
replaced before the drawing on trial N + 1.
oIf we made a very large number of draws, we would find that (very nearly) 24%
of the draws would result in a blue M&M. Thus we might define probability as
the limit of the relative frequency of occurrence of the desired event that we
approach as the number of draws increases.
-Subjective probability: Definition of probability in terms of personal subjective belief in
the likelihood of an outcome.
oExample: “I think that tomorrow will be a good day,” is a subjective statement of
degree of belief, which probably has very little to do with the long-range relative
frequency of the occurrence of good days
oSubjective probabilities play an extremely important role in human decision-
making and govern all aspects of our behavior.
5.2 – Basic Terminology Rules
-Event: a term that statisticians use to cover just about anything
oThe occurrence of a king when we deal from a deck of cards, a score of 36 on a
scale of likeability, a classification of female for the next person appointed to
Supreme Court, the mean of a sample
oThe probability of “something”… that “something”= an event
-Independent events: when the occurrence or nonoccurrence of one has no effect on the
occurrence or nonoccurrence of the other
oThe voting behaviors of two randomly chosen subjects normally would be
assumed to be independent, especially with a secret ballot, because how one
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Chapter #5 Lec 4
person votes could not be expected to influence how the other will vote.
othe voting behaviors of two members of the same family probably would not be
independent events, because those people share many of the same beliefs and
attitudes.
- Two events are said to be mutually exclusive if the occurrence of one event precludes
the occurrence of the other.
oFor example, the standard college classes of First Year, Sophomore, Junior, and
Senior are mutually exclusive because one person cannot be a member of more
than one class.
- A set of events is said to be exhaustive if it includes all possible outcomes.
oThus the four college classes in the previous example are exhaustive with respect
to full-time undergraduates, who have to fall in one or another of those
categories—if only to please the registrar’s office.
oAt the same time, they are not exhaustive with respect to total university
enrollments, which include graduate students, medical students,
nonmatriculated students, hangers-on, and so forth.
Basic Laws of Probability
- Two important theorems are central to any discussion of probability referred to as the
-additive law of probability: Given a set of mutually exclusive events, the probability of
the occurrence of one event or another is equal to the sum of their separate
probabilities.
oExample: M&Ms
probability that p(blue) = 24/100 = .24, p(green) = 16/100 = .16, and so
on.
Additive law: Thus, p(blue or green) = p(blue) + p(green) = .24 + .16 = .40.
The occurrence of one event precludes the occurrence of the other
If an M&M is blue, it can’t be green
Multiplicative Rule
-Multiplicative law of probability: The probability of the joint occurrence of two or more
independent events is the product of their individual probabilities.
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Chapter #5 Lec 4
oExample: M&Ms
probability that p(blue) = 24/100 = .24, p(green) = 16/100 = .16, and
p(other) = .60
oMultiplicative law: Thus p(blue, blue) = p(blue) X p(blue) = .24 X .24 = .0576.
oSimilarly, the probability of a blue M&M followed by a green one is p(blue,
green) = p(blue) X p(green) = .24 X .16 = .0384.
Notice that we have restricted ourselves to independent events, meaning
the occurrence of one event can have no effect on the occurrence or
nonoccurrence of the other.
- Example: Race and death sentence
oWe need to calculate what this probability would be if the two events (race and
death sentence) are independent, as would be the case if verdicts are race-blind.
o If we assume that these two events are independent, the multiplicative law tells
us that p(nonwhite, death) = p(nonwhite) X p(death).
oIn their study 34.4% of the defendants were nonwhite, so the probability that a
defendant chosen at random would be nonwhite is .344.
oSimilarly, 8% of the defendants received a death sen- tence, giving p(death) = .
08.
oTherefore, if the two events are independent, p(nonwhite, death) = .344 X .08 = .
028 = 2.8%.
- Using both the additive law and multiplicative law:
oWhat is the probability that I will draw one blue M&M and one green M&M?
1st use the multiplicative rule ->
2nd use the additive rule ->
oThus the probability of obtaining one M&M of each of those colors over two
draws is approximately .08
- When to use the rules:
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