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Chapter 3- Probability
3.1
Experiment: An act or process that generates
well-defined outcomes.
Example:
1. Toss a coin.
2. Roll a die.
3. Selecting a random sample of
size 2 from a group of five.
Sample Space: set of all possible outcomes of
an experiment.
Simple Event: An individual outcome to an
experiment.
Event: a subset (part) of the sample space.
Now we wish to assign probabilities to
experimental outcomes. There are three
approaches that are used most frequently.
1. The Classical Approach
2. The Relative Frequency Approach
3. The Subjective Approach
Regardless of the method used, the probabilities
assigned must satisfy two basic requirements:
1. The probability assigned to each
experimental outcome E must be
between 0 and 1. That is,
i
0  P(E )  1
i
2. If S = {E , E , …, E n }, then
2
1
P(E ) +P(E ) + …+ P(E n ) = P(S) =1
1
2
Note: The probability of an event is sum of the
probabilities of the simple events which
comprise it.
Classical Approach: Assume that a given
experiment has n different outcomes, each of
which has an equal chance of occurring. If the
event E can occur in m of these n ways, then
Probability of E = P(E) = m/n.
Examples:
1. Toss a six-sided die. List the elements of the
sample space and find
a. P(“Odd number”)
b. P(“Number > 4”)
2. Toss two coins. List the elements of the
sample space and find P(“At least one
head”).
3. Toss three coins. List the elements of the
sample space and find the probability of the
following events:
a. A: “(exactly) two heads”
b. B: “at least two heads”
c. C: “at most two heads”
Soln: P(A) = 3/8, P(B) = 4/8, P(C) = 7/8
4. Roll two dice. Find the probability of the
following events:
a. A: “Sum = 7”
b. B: “Sum = 11”
c. C: “Sum = 12”
d. D: “Sum < 5”
Soln: P(A) = 6/36, P(B) = 2/36, P(C) = 1/36,
P(D) = 6/36 .
Sum
2
3
4
5
6
7
8
9
10
11
12
Prob.
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Relative Frequency Approach: The relative
frequency approach of assigning probabilities is
appropriate when the data are available to
estimate the proportion of the time the
“outcome” will occur when the experiment is
repeated a large number of times. Note that this
approach does not require that each
experimental outcome is equally likely.
Method for approximating P(E): Conduct (or
observe) an experiment a large number of times
and count the # of times the event E actually
took place.
P(E) 
# of times E occurred
# of times exp eriment was repeated
Note: As the experiment is repeated again and
again, the empirical probability of success tend
to approach the actual probability.
Example 5 The final exam in a course resulted in
the following grades
Grade
Number
A B C D F
7 12 16 5 3
a. What is the probability that a randomly
selected student received an A?
b. What is the probability that a randomly
selected student received a C?
Example 6: Faced with the question of
determining the probability of obtaining either 0
heads, 1 head, or 2 heads when flipping a coin
twice, an individual argued that it seems
reasonable to treat the outcomes equally likely
and that the probability of each event is 1/3. Do
you agree?
Example 7: The Swiss astronomer Wolf tossed
two dice 100,000 times and observed that pairs
came up 16,466 times. (a) Use these data to
estimate the estimate the probability of rolling a
pair. (b) What is actual probability of getting a
pair when we roll two dice?
Example 8: Tom has signed up for MATH 2620.
Based on his past record, he estimates that there
is 0.1 probability that he will withdraw from the
course and that he is twice as likely to pass the
course than to fail the course. What is Tom’s
probability of passing this course?
Subjective Method – Discuss in Class
Homework: 1-10 (all), 13-21 (odd) pp. 117-118
3.3 Rules for Computing Event Probabilities:
Venn Diagrams
Union
The union of two events A and B is the
event containing all sample points in A or B
or both. Notation: A  B.
Intersection
The intersection of two events A and B is
the event composed of all sample points that
are in both A and B. Notation: A  B.
Note:
P(A  B) = P(A or B) = P(event A occurs or
event B occurs or they both occur)
P(A  B) = P(A and B) = P(event A and B both
occur)
Addition Rule:
P(A  B) = P(A) + P(B) – P(A  B)
Example 9: In a study of 100 students that had
been awarded university scholarships, it was
found that 40 had part-time jobs, 25 had made
the dean’s list the previous semester, and 15 had
both a part-time job and had made the dean’s
list. What was the probability that a student had
a part-time job or was on the dean’s list?
Soln: 0.5
Example 10: You are playing a card game in
which spades and honors (Ace, Queen, King, or
Jack) are valuable. If you draw a card from the
full deck, what is the probability that it is a
“valuable card?”
Soln: 25/52
Mutually Exclusive Events: (“ME”) Two
events A and B are called mutually exclusive if
A  B contains no sample points. That is, A and
B have no outcomes in common.
Note: If A and B are mutually exclusive, then
P(A  B) = 0.
Examples of “ME” Events
Testing a subject with IQ > 100
Testing a subject with IQ < 100
Selecting a voter who is a registered Democrat
Selecting a voter who is a registered Republican
Examples of Events that are not “ME”
Selecting a doctor who is a brain surgeon
Selecting a doctor who is a woman
Selecting a voter who is registered Democrat
Selecting a voter who is under 30 years of age
Addition Rule for “ME” Events
If A and B are “ME”, then
P(A  B) = P(A) + P(B)
Example 11: If one card is drawn from a full
deck of 52 cards, what is the probability that the
card drawn is a queen or a six?
Soln. 2/13
Example12: If you roll a pair of dice, what the
probability of getting a one on the first die or the
second die?
Soln: 11/36
Complement of an Event: If A is an event over
the sample space S, the complement of A
(Notation: A’, A , A ) is defined to be the event
consisting of all sample points in S that are not
in A.
C
Subtraction Rule: P(A) + P(A ) = 1
That is,
P(A ) = 1 – P(A) or P(A) = 1 - P(A )
C
C
C
Example 13: Let
H = person has high blood person,
E = person never exercises at all,
S = person never smoked, and
P(H) = 0.12, P(E) = 0.25, P(S) = 0.46
(a) Are any of these events mutually exclusive?
(b) What is the probability that a randomly
selected person has smoked?
(c) Is the probability that a person never
exercises at all or has high blood person >
0.4? Explain.
Soln. (a) No (b) 0.54 (c) No
Homework: 1-10 (all), 13-19* (odd)
pp. 135-137
*: Do # 19 after completing Section 3.2
3.2 Conditional Probability
P(A|B) = P(A  B)/P(B)
P(B|A) = P(A  B)/P(A)
Example 14: Roll a die. Let A = an even number
shows up, and B = a number > 3 shows up. Find
(a) P(A) (b) P(B) (c) P(A  B) (d) P(A|B)
Soln. (a) ½ (b) ½ (c) 1/3 (d) 2/3
Note: In general, P(A)  P(A|B)
It is important to recognize the difference
between P(A) and P(A|B). The expression P(A)
means “considering all possible outcomes, what
is the probability that A occurs?” The expression
P(A|B) means “assuming that B has occurred,
what is the probability that A also occurs?”
Independent Events Two events A and B are
called independent if the occurrence of one does
not affect the probability of the occurrence of
the other. That is,
P(A|B) = P(A) and P(B|A) = P(B)
Note: P(A|B) = P(A) if and only if
P(B|A) = P(B). Thus to show that events A and
B are independent, it is enough to show that
P(A|B) = P(A) or P(B|A) = P(B).
Example 16: Draw a card from a deck of
52cards. Let A = a red card is drawn, and B = a
jack is drawn.
(a) Are A and B independent?
(b) Are A and B mutually exclusive?
Soln. (a) yes (b) no
Multiplication Rule
(a) When the events are not independent
P(A  B) = P(A)P(B|A)
P(A  B) = P(B)P(A|B)
(b) When the events are independent
P(A  B) = P(A)P(B)
Theorem: A and B are independent if and only if
P(A  B) = P(A)P(B)
Example 17: Let A and B be events of the
sample space S such that P(A) = 0.7, P(B) = 0.5,
P(A  B) = 0.35.
(a) Are A and B mutually exclusive?
(b) Are A and B independent?
(c) Find P(A  B)
(d) Find P(A  B )
(e) Find P(B  A )
Example 18: #10 on page 126
c
c
Example 19: #14 on page 127
Example 20: #18 on page 128
Homework: 1-8(all), 11, 13-17 (odd)