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Chapter 5

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

by OC648217

Department

PsychologyCourse Code

PSYB07H3Professor

Dwayne PareChapter

5 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-

ment.

-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)

1

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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 individual’s 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.

3

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###### Document Summary

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. 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.

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