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Chapter 5

PSYB07H3 Chapter Notes - Chapter 5: Mutual Exclusivity, Seat Belt, Continuous Or Discrete Variable

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Dwayne Pare

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PSYBO7 - Chapter 5: Basic Concepts of Probability
- The material covered in this chapter has been selected for two reasons.
-First, it is directly applicable to an understanding of the material presented in the remain-
der of the book.
-Second, it is intended to allow you to make simple calculations of probabilities that are
likely to be useful to you. Material that does not satisfy either of these qualifications has
been deliberately omitted.
5.1 Probability
- The concept of probability can be viewed in several different ways
-There is not even general agreement as to what we mean by the word probability.
-The oldest and perhaps most common defini-
tion of a probability is called the analytic view
-Analytical View of Probability: Defini-
tion of probability in terms of analysis of
possible outcomes.
-One of the examples that is often drawn
into discussions of probability is that of
one of my favorite candies, M&M’s.
M&M’s are a good example because ev-
eryone is familiar with them, they are
easy to use in class demonstrations. The
Company keep lists of the percentage of colors in each bag—though they seem to keep moving
the lists around, making it a challenge to find them on occasions. At present the data on the milk
chocolate version is shown in Table 5.1
-Suppose that you have a bag of M&M’s in front of you and you reach in and pull one out. Just to
simplify what follows, assume that there are 100 M&M’s in the bag, though that is not a require-
-What is the probability that you will pull out a blue M&M? You can probably answer this ques-
tion without knowing anything more about probability.
-Because 24% of the M&M’s are blue, and because you are sampling randomly, the proba-
bility of drawing a blue M&M is .24.
-This example illustrates one definition of probability: If 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 oc-
currence is A /(A 1 B ), and the probability of its failing to occur is B /(A 1 B ).
-Because 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 = 24, B
= 76, and p(A ) = 24/(24 + 76) = .24.
- An alternative view of probability is the frequentist view
-Relative Frequentist View: Definition of probability in terms of past performance
-Suppose that we keep drawing M&M’s from the bag, noting the color on each draw. In conduct-
ing this sampling study we sample with replacement (Sampling in which the item drawn on trial
N is replaced before the drawing on trial N + 1.), meaning that each M&M is replaced before the
next one is drawn. If we made a very large number of draws, we would find that (very nearly)
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PSYBO7 - Chapter 5: Basic Concepts of Probability
24% of the draws would result in a blue M&M. Thus we might define probability as the limit1 of
the relative frequency of occurrence of the desired event that we approach as the number of
draws increases.
- The third concept of probability is advocated by a number of theorists. That is the concept of subjec-
tive probability
-Subjective Probability: Definition of probability in terms of personal subjective belief in the
likelihood of an outcome.
- By this definition probability represents an individuals subjective belief in the likelihood of the
occurrence of an event. For example, the statement, “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, and in fact may have no mathematical
basis whatsoever
-Subjective probabilities play an extremely important role in human decision-making and govern
all aspects of our behavior
-Statistical decisions as we will make them have generally been stated with respect to frequentist
or analytical approaches, although even so the interpretation of those probabilities has a strong
subjective component. Recently there seems to have been a shift toward viewing probabilities
more subjectively
-Ex: Bayes’ Theorem which is essentially the use to subjective possibilities
- Any of the definitions will lead to essentially the same result in terms of hypothesis testing
-In actual fact most people use the different approaches interchangeably
5.2 Basic Terminology
- The basic bit of data for a probability theorist is called an event
-Event: the outcome of a trial
-The word event is a term that statisticians use to cover just about anything
-Whenever you speak of the probability of something, the “something” is called an event
-Ex: When we are dealing with a process as simple as flipping a coin, the event is the out-
come of that flip—either heads or tails. When we draw M&M’s out of a bag, the possible
events are the 6 possible colors. When we speak of a grade in a course, the possible events
are the letters A, B, C, D, and F.
- Two events are said to be independent events when the occurrence or nonoccurrence of one has no ef-
fect on the occurrence or nonoccurrence of the other
-Independent Events: Events are independent when the occurrence of one has no effect on the
probability of the occurrence of the other.
-Ex: The voting behaviors of two randomly chosen subjects normally would be assumed to be in-
dependent, especially with a secret ballot, because how one person votes could not be expected to
influence how the other will vote.
- Two events are said to be mutually exclusive if the occurrence of one event precludes the occurrence
of the other
-Mutually Exclusive: Two events are mutually exclusive when the occurrence of one precludes
the occurrence of the other
-Ex: the standard college classes of First Year, Sophomore, Junior, and Senior are mutually exclu-
sive because one person cannot be a member of more than one class
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PSYBO7 - Chapter 5: Basic Concepts of Probability
-A set of events is said to be exhaustive if it includes all possible outcomes.
-Exhaustive: A set of events that represents all possible outcomes.
-Ex: Thus 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. At the same time, they are not exhaustive with respect to total
university enrollments, which include graduate students, medical students, non matriculated
students, hangers-on, and so forth.
- Probabilities range between 0.00 and 1.00.
-If some event has a probability of 1.00, then it must occur ( though very few things have a proba-
bility of 1.00)
-If some event has a probability of 0.00, it is certain not to occur.
-The closer the probability comes to either extreme, the more likely or unlikely is the occurrence
of the event
Basic Laws of Probability
- Two important theorems are central to any discussion of probability
-They are often referred to as the additive and multiplicative rules.
The Additive Rule
- Additive Law of Probability: The rule giving the probability of the occurrence of one or more mutu-
ally exclusive events
-Ex: To illustrate the additive rule, we will use our M&M’s example and consider all six colors.
From Table 5.1 we know from the analytic definition of probability that p (blue)= 24/100 = .24,
p (green) = 16/100 = .16, and so on .
-But what is the probability that I will draw a blue or green M&M instead of an M&M of
some other color?
- *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.
-Thus, p (blue or green) = p (blue) + p (green) = .24 + .16 = .40
-Notice that we have imposed the restriction that the events must be mutually exclu-
sive, meaning that the occurrence of one event precludes the occurrence of the other. If
an M&M is blue, it can’t be green. This requirement is important.
-Ex: About one-half of the population of this country are female, and about one-half of the population
is taller than 5'6.".2 But the probability that a person chosen at random will be female or will be taller
than 5'6." is obviously not .50 + .50 = 1.00. Here the two events are not mutually exclusive.
-Ex: The probability that a girl born in Vermont in 1987 was named Ashley or Sarah, the two most
common girls’ names of that year, equals p (Ashley) + p (Sarah) = .010 + .009 = .019. Here the
names are mutually exclusive because you can’t have both Ashley and Sarah as your first name
The Multiplicative Rule
-Multiplicative Law of Probability: The rule giving the probability of the joint occurrence of indepen-
dent events
-Ex: Let’s continue with the M&M’s where p (blue) = .24, p (green) = .16, and p (other) = .60.
Suppose I draw two M&M’s, replacing the first before drawing the second.
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