This document discusses probability theory and concepts such as classical probability, frequency distributions, and rules of probability including addition, multiplication, and conditional probability. It provides examples of how to calculate probabilities for events like coin tosses or sampling from populations. The key points are: - Classical probability examines the likelihood of events based on the number of possible outcomes. It ranges from 0 to 1. - The addition rule is used to calculate the probability of events occurring together or separately. It adds probabilities minus any joint probability. - The multiplication rule is used for independent events and calculates the probability of sequences by multiplying individual event probabilities. - Conditional probability examines the likelihood of one event given that another event occurred.

1. PROBABILITY
THEORY
GAMES OF CHANCES
By DesmondAyim-Aboagye, PhD

2. Probability
■ the relative frequency with which some event actually occurs or
is likely to occur again in the future.

3. Examples:
■ Car Insurance: Probability of car getting accident
■ Health Insurance: Probability of getting sick
■ Marriage and divorce: 50: 50

4. Statisticians who study probability
theory and behavioral scientists who rely on
probabilistic concepts share an interest in making
quantitative statements about the likelihood or
occurrence of uncertain events.
We already quantified aspects of uncertainty in
different ways earlier in this book.

5. * we discussed how linear regression is used to
make particular predictions that are often uncertain (e.g., Will
information collected at
one time estimate future performance?).
* Similarly, a correlation indicates whether the
relationship between two variables is positive, negative, or
zero, but it does not specify
the causes of the relationship (see chapter 6)-such factors
remain unknown unless an
experimental investigation is undertaken.
* Even the results of such experiments remain
uncertain until independent variables are manipulated,
dependent measures are
measured, and data are collected and appropriately analyzed.

6. Formal
properties of probability theory
■ probability entails examining the ratio of the number of
actual occurrences of some event to the total number of
possible occurrences for the event.
■ The link shared between samples and populations is often
described in probabilistic terms: how likely is that sample A
was drawn from population B rather than population C?

7. When a sample is established as originating from a given population, empirical research and
statistical analysis can be used to make inferences about the characteristics of the
population.
These inferences are based on inferential statistics, which enable investigators
to use the limited information from a sample to draw broad conclusions
about a population.

8. The Gambler's Fallacy and Randomness
Revisited
■ Tossing a Coin = it produces a Head (H) orTail (T)
■ 50:50 chance that it will be aT or H.
■ Consider this sequence:
■ H-H-H-T-T-T
■ H-T-H-T-T-H
■ Neither one is more likely than the other, but many people will nonetheless select the second array
because it appears to be relatively more "random" than the first one.
■ The first is too ORDERED/SEQUENCE, while the second is RANDOM.

9. Coins obey the law of averages, meaning that if a coin is
fair, there is an equal chance that the next flip will result
in a head or a tail (Runyon, Haber, Pittenger, &
Coleman, 1996).
Statisticians note that in the case of coin tosses or
other random events, each event is independent of
every other event.
An Important Rule!!!

10. INDEPENDENCE
Independence exists when the probability of one
event is not affected by the probability of any
previous events.
Put more simply still, coins have
no memory-only coin flippers do.

11. Probability: ATheory of Outcomes
■ Probability is like the cane that the blind man uses to feel his
way. If he could see, he would not need the cane, and if I knew
which horse was the fastest, I would not need probability
theory.
■ --STANISLAW LEM

12. CLASSICAL PROBABILITYTHEORY:
CONCEPTS
■ When we speak of the classical approach to probability, we refer to a mathematical
account of the likelihood that one event will occur in the context of all other, possible
events. In this case, probability is usually determined by the frequency with which the
event of interest and the other events can occur within a sample space.
■ A sample space is the frame of reference for any probability, as it is defined as all the
potential outcomes that can be observed or measured.

13. Examples: Sample space
■ The students in my statistics class comprise not only a sample but also a sample space.
■ When I call on anyone of them, I am sampling their opinion-one student is an
observation from the larger sample space of students.
■ Sample space (A) occurring and (not A) occurring
■ Symbolically p (A) [probability of event A]

14. Probability (A)
■ P (A) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑖𝑛𝑔 𝑒𝑣𝑒𝑛𝑡 𝐴
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
■ Conceptually equivalent to:
■ P (A) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑖𝑛𝑔 𝑒𝑣𝑒𝑛𝑡 𝐴
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑖𝑛𝑔 𝐴 + 𝑇ℎ𝑜𝑠𝑒 𝑛𝑜𝑡 𝑓𝑎𝑣𝑜𝑢𝑟𝑖𝑛𝑔 𝐴

15. Probability of interest in any sample
space as ''A.'
■ Number of outcomes = known frequencies of events which can be defined as:
■ f (A), that is, “for the frequency of A” and f (not A), that is, “the frequency of not A”
■ Symbolically: P (A) =
𝒇( 𝑨)
𝒇 𝑨 +𝒇 (𝒏𝒐𝒕 𝑨)

16. Sampling with Replacement. What is the probability that you will draw a red marble?We
know there are 6 red marbles and 10 black ones. If we designate the red marbles "X' and the black marbles
"not A:' then based on [8.3.1]' the probability of A can be determined by:
[8.3.2] P(A) =
𝟔
𝟔+𝟏𝟎
[8.3.3) P(A) =
𝟔
𝟏𝟔
[8.3.4) P (A) =.375
Thus, the probability of event A, the selection of one red marble from the population of
16 marbles, is .375.We will explain how to interpret this numerical probability below let's
continue to focus on calculation for a bit longer.
Mathematical Example

17. What about the probability of selecting a black marble?
8.4.1. P(not A) =
𝑷(𝒏𝒐𝒕 𝑨)
𝑷 𝒏𝒐𝒕 𝑨 +𝑷(𝑨)
8.4.2 P(not A) =
𝟏𝟎
𝟔+𝟏𝟎
8.4.3 P(not A) =
𝟏𝟎
𝟏𝟔
8.4.4 P (not A) = .625

18. Sample with replacement
■ Here is an important lesson we can point to now and come back to later:
■ [8.5.1] p(A) + p(not A) = 1.00.
■ The probability of selecting a red marble plus the probability of selecting a black
marble is equal to 1.00, or:
■ [8.5.2] .375 + .625 = 1.00.

19. Sample without replacement
■ That is, if we do not replace the sample after having drawn 1, it will remain 5 (no longer
6).The total will be then 15 (no longer 16). Reworking with these figures, we shall get
the probability ofA,
■ P (A) = p(A) = .333.
■ P (not A)= .667
■ Naturally, the two probabilities-p(A) + p(not A)-still sum to 1.00
■ (Le., .333 + .667).

20. Probability (important)
■ Probability refers to an event that is likely to happen, not that
event must happen. Higher probabilities indicate what events
are apt to take place in the long run.

21. Classical probability: A few key points
■ 1. Any probability discloses a pattern of behavior that is expected to occur in the long run.
■ 2. A given probability can take on a value ranging from 0 to 1.00.
■ 3. Probabilities can but need not be rounded. If a probability is determined to be, say,
■ P(A) = .267, it can be reported as .267 or rounded to .27. For pedagogical purposes
■ and issues of calculation accuracy, the convention in this book is to report a probability
■ to three places behind the decimal point. Be advised that the convention of the
■ American Psychological Association (1994) is to report such information to only two
■ places behind a decimal point.

22. Probabilities Can Be Obtained from Frequency Distributions
X f
10
8
6
5
3
2
2
4
3
7
4
5
∑f = 25
We can determine the simple probability of any event X within a frequency distribution by
using this formula:
8.8.1 p (X) = f/N
8.8.2. p (x =10) = 2/25 = .080
Probabilities Can Be Obtained from Frequency Distributions

23. Besides determining the probability of any single event X, we can also explore the
probability associated with sampling a range of possible values within a population. In
other words, if we randomly select an observation X from the above frequency
distribution, what is the probability it will be less than «) 6? If you look at the above
frequency table, you can see that X is less than 6 when X is equal to 5, 3, or 2.We need
only sum the frequencies (fs) associated with these three observations (X) to determine
the probability of selecting an observation less than 6, or: 16
[8.9.1] p(X < 6) = 16/25 = .640.
This probability was determined by summing the fs of 7,
4, and 5, which correspond to
X = 5, 3, and 2, respectively.
Probabilities Can Be Obtained from Frequency Distributions

24. The Addition Rule for Mutually Exclusive
and Non-mutually Exclusive Events
■ The addition rule for probability is used when considering the "union"or intersection of
two events.
■ The addition rule for probability indicates that the probability of the union of two
events, A and B, that is, p(A or B), is equal to: p(A or B) = p(A) + p(B) - p(A and B).

25. Imagine, for example, that we have a small class of 20 elementary school
students.
Eight of the children have blond hair, 6 have blue eyes, and 4 have both
blonde hair and blue eyes.What is the probability of selecting a child
who has blond hair or blue eyes? (Not both.) In evaluating this sample,
we notice a few things. First, we are considering a subset of the children,
not all 20, though we will use 20 as the denominator for our probability
calculations. Second, we know the probability of having blond hair is
equal to 8/20 (i.e., p(A) = 0400) and that associated with blue eyes is
equal to 6/20 (i.e., (B) = .300).Third, we know that a joint probability-
having both blond hair and blue eyes-is also present.
Addition Probability

26. Joint Probability
■ A joint probability is the mathematical likelihood of selecting an
observation wherein two conditions- p(A) and p(B)-are present. In
formula terms, a joint probability is equal to p(A and B).
■ "What is the probability of selecting a child from the sample who
has blond hair and blue eyes? "we could readily answer it using the
information presented: p (A and B) = 4/20 = .20.

27. Chapter 8 Probability
P (A and B)
A B
Figure 8.2 Venn Diagram of Sample of Elementary School Children
Note:The star-shaded area indicates the joint probability (i.e., p (A and B)), the overlap
between p(A) and p (B) representing the four children in the sample who have both blond hair
and blue eyes.
A B

28. The probability of selecting a blond-haired or a blue-eyed child from
the sample (and not a blond haired, blue eyed child), then, would be:
[8.12.1] p(A or B) = p(A) + p(B) – p(A and B),
[8.12.2] p(A or B) = (0400 + .300) - .200,
[8.12.3]p(A or B) = .700 - .200,
8.12.4] p(A or B) = .500.

29. Mutually Exclusive.
■ As you can see, the probabilities of A and B are added
together, and then the joint probability of A and B is
subtracted from the resulting sum.Thus, when randomly
selecting a child from the sample, we know that the
probability is .50 that the child will have either blond hair or
blue eyes.
■ The addition rule is also applicable to probabilistic events that
are said to be mutually exclusive.

30. Two events are mutually
exclusive when they have no
observations within a sample in
common.
Mutually exclusive events cannot
occur simultaneously.
Specifically, the joint probability of A
and B, that is,
P(A and B), is equal to O.

31. Figure 8.3Venn Diagram of Second Sample of Elementary School Children
Note:The probabilities of A and B are said to be mutually exclusive; that is, they share not
overlapping space with one another. No children in the sample have both blond hair and blue
eyes (i.e.., p(A and B) = 0).
A B

32. The Multiplication Rule for Independent
and Conditional Probabilities
■ What is the probability of flipping H-T -H-H-T?.
■ In this case, [we refer] a sequence of events refers to joint or co-occurrence of two or even
more events, such as a particular pattern of coin tosses.
■ Determining the probability of a precise sequence of coin flips involves recognizing the
independence of each flip from every other flip.
■ To determine this probability, we rely on the multiplication rule for independent events.

33. “When a sequence of events is independent, the
multiplication rule entails multiplying the probability of
one event by the probability of the next event in the
sequence, or p(A then B then· … ) = p(A) x p(B) x p (...).”
Multiplication Rule

34. If we want to know the probability of a sequence of coin flips, such as the aforementioned
pattern H-T -H-H-T, we need only remember that the probability of an H or aT on any toss
is .500. [1/2=.500]
[8.14.11] p(H thenT then H then H thenT) = p(H) X p(T) X p(H) X p(H)
X p(T),
[8.14.21] p(H thenT then H then H thenT) = (.500) X (.500) X (.500)
X (.500) X (.500),
[8.14.31] p(H thenT then H then H thenT) = .031.
Example:

35. CONDITIONAL PROBABILITY
■ A conditional probability exists when the probability
of one event is conditional-that is, it depends on-the
role of another event.

36. Gender High Self-
Monitors
A1
Low-Self
Monitors
A2
Marginal
Probability
A
Male, B1
Female, B2
Marginal
probability, B
.367
.167
.533
.153
.313
.467
.520
.480
1.00
Table 8.2 Joint and Marginal Probabilities of Students by Self-MonitoringType and
Gender
Table 8.2 contains two important types of probability information. First, the four
entries in the center of the table are joint probabilities shared between the two levels of A (self-
monitoring type) and B (gender).The probability of being a high self-monitor
and a male (p(AI and BI )), for example, is .367, just as the relative likelihood of being a
low self-monitor and a female (p(A2 and B2)) is.313 (seeTable 8.2). Please note that the
sum of the four joint probability cells must sum to 1.00 (i.e., .367 + .153 + .167 + .313
= 1.00). It is always a good idea to perform this quick check for calculation errors before using or
reporting the probabilities.

37. Marginal Probability
■ A marginal probability, sometimes known as an
"unconditional probability," indicates the likelihood of
an independent event's occurrence.

38. Table 8.2 also includes what are called marginal probabilities, probabilities based on
collapsing across one of the two variables shown in the table. For example, by
glancing down columnsAl and A2, the probability of being a high or a low self-
monitor, respectively,
can be known. In the same way, reading across the two rows representing gender will
reveal the probability of being a male or a female in the sample.
As shown in the far right ofTable 8.2, the marginal probability of being a male in the
sample (p(B j )) is .520. Similarly, the marginal probability of being a high self-
monitor(p(A j ) = .533) is slightly higher than the probability of being a low self-
monitor(p(A2 ) = .467; seeTable 8.2).The sum of the marginal probabilities for either
personality
type (A1 + A2 ) or gender (B1 + B2 ) must also sum to 1.00 (seeTable 8.2). As noted
above, it is always appropriate to determine that each respective set of marginal
probabilities do sum to 1.00 to eliminate any errors that could plague later
calculations.

39. Although they are related, conditional
probabilities differ from joint probabilities
and marginal probabilities. Conditional
probabilities incorporate joint and
marginal probabilities.

40. The probability of being a high self-monitor (p(A 1 ))
given he is a male (p(B 1 )), or
P (A1│B1) =
P (A1and B2)
𝑃 (𝐵1)

41. Thus, we will use the joint probability of being a high self-monitor and
male (p(A 1 and B2)). If we take these probabilities from Table 8.2 and
enter them into [8.16.1], we get
[8.18.2] P (A1│B1) =
.367
.520
[8.18.3) P (A1│B1) = .706.