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```Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola
Section 4.2-1
Chapter 4
Probability
4-1 Review and Preview
4-2 Basic Concepts of Probability
4-4 Multiplication Rule: Basics
4-5 Multiplication Rule: Complements and Conditional
Probability
4-6 Counting
4-7 Probabilities Through Simulations
4-8 Bayes’ Theorem
Section 4.2-2
Key Concept
This section presents three approaches to
finding the probability of an event.
The most important objective of this
section is to learn how to interpret
probability values.
Section 4.2-3
Definitions

Event
any collection of results or outcomes of a procedure

Simple Event
an outcome or an event that cannot be further broken
down into simpler components

Sample Space
for a procedure consists of all possible simple
events; that is, the sample space consists of all
outcomes that cannot be broken down any further
Section 4.2-4
Example
In the following display, we use “b” to denote a
baby boy and “g” to denote a baby girl.
Procedure
Example of
Event
Sample Space
Single birth
1 girl (simple
event)
{b, g}
3 births
2 boys and 1 girl
(bbg, bgb, and
gbb are all simple
events)
{bbb, bbg, bgb,
bgg, gbb, gbg,
ggb, ggg}
Section 4.2-5
Notation for
Probabilities
P - denotes a probability.
A, B, and C - denote specific events.
P(A) - denotes the probability of
event A occurring.
Section 4.2-6
Basic Rules for
Computing Probability
Rule 1: Relative Frequency Approximation of
Probability
Conduct (or observe) a procedure, and count the
number of times event A actually occurs. Based on
these actual results, P(A) is approximated as
follows:
P(A) =
# of times A occurred
# of times procedure was repeated
Section 4.2-7
Example – Relative Frequency Concept
of Probability
Example #9:
A sample of 820 adults showed that 80 of them had no credit cards, 116 had one card
each, 94 had two cards each, 77 had three cards each, 43 had four cards each, and
410 had five or more cards each. Write the frequency distribution table for the
number of credit cards an adult possesses. Calculate the relative frequencies for all
categories. Suppose one adult is randomly selected from these 820 adults. Find the
(a) three credit cards (b) five or more cards
Solution:
# of Credit
Cards
0
1
2
3
4
>=5
Frequency
(f)
80
116
94
77
43
410
∑f = 820
Relative
Frequency (rf)
80/820 = 0.0976
116/820 = 0.1415
94/820 = 0.1146
77/820 = 0.0939
43/820 = 0.0524
410/820 = 0.5000
∑rf = 1.0000
a. P(3 cards) = 0.0939
b. P(>=5) = 0.5000
Section 4.2-8
Basic Rules for
Computing Probability
Rule 2: Classical Approach to Probability
(Requires Equally Likely Outcomes)
Assume that a given procedure has n different
simple events and that each of those simple events
has an equal chance of occurring. If event A can
occur in s of these n ways, then
s
number of ways A can occur
P ( A) = =
n number of different simple events
Section 4.2-9
Example – Classical Probability
Example #6:
Solution:
A box contains 40 marbles. Of them,
18 are red and 22 are green. If one
# of events favorable to red 18
a.
P
(
red
)


 0.45
marble is randomly selected out of this
Total # of outcomes
40
box, what is the probability that this
# of events favorable to green 22
marble is:
b. P(green ) 

 0.55
Total # of outcomes
40
a. red?
b. green?
Example #7
A multiple-choice question on a test
has five answers. If Dianne chooses
one answer based on “pure guess” ,
what is the probability that her answer
is:
a. correct?
b. wrong?
Do these two probabilities add up to
1? If yes why?
Solution:
# of events favorable to correct 1
  0.2
Total # of outcomes
5
# of events favorable to wrong 4
b. P( wrong ) 
  0.8
Total # of outcomes
5
a. P(correct ) 
Yes. The experiment has two and only two
outcomes and according to the 2nd property of
probability, the sum of the probability must be 1.
Section 4.2-10
Example – Classical Probability
Example #5:
Suppose a randomly selected
passenger is about to go through the
metal detector at JFK Airport in New
York City. Consider the following two
outcomes.
The passenger sets off the metal
detector, or the passenger does not
set off the metal detector. Are the two
outcomes equally likely? Explain why
or why not.
If you are to find the probability of
these two outcomes, would you use
the classical approach or another
approach? Explain why.
Solution:
a. The two outcomes, “passenger sets off the metal
detector” and “passenger does not set of the metal
detector”, are not equally likely because if they
were, 50% of the passengers would set off the
detector. This would be a daunting task for the
b.Classical approach will not be appropriate for
determining the probability of these two outcomes.
Therefore, another approach is needed. This other
approach will require obtaining a random sample of
passengers going through NY JFK airport and
collecting information whether they set off the
detector or not.
Section 4.2-11
Basic Rules for
Computing Probability
Rule 3: Subjective Probabilities
P(A), the probability of event A, is estimated by
using knowledge of the relevant circumstances.
Section 4.2-12
Law of Large Numbers
As a procedure is repeated again and again, the
relative frequency probability of an event tends to
approach the actual probability.
Section 4.2-13
Example
When three children are born, the sample space is:
{bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}
Assuming that boys and girls are equally likely, find
the probability of getting three children of all the
same gender.
2
P  three children of the same gender    0.25
8
Section 4.2-14
Simulations
A simulation of a procedure is a process that
behaves in the same ways as the procedure
itself so that similar results are produced.
Section 4.2-15
Probability Limits
Always express a probability as a fraction or
decimal number between 0 and 1.
 The probability of an impossible event is 0.
 The probability of an event that is certain to
occur is 1.
 For any event A, the probability of A is
between 0 and 1 inclusive.
That is, 0  P( A)  1 .
Section 4.2-16
Possible Values
for Probabilities
Section 4.2-17
Complementary Events
The complement of event A, denoted by
A, consists of all outcomes in which the
event A does not occur.
Section 4.2-18
Example
1010 United States adults were surveyed and 202
of them were smokers.
It follows that:
202
P  smoker  
 0.200
1010
202
P  not a smoker   1 
 0.800
1010
Section 4.2-19
Rounding Off Probabilities
When expressing the value of a probability, either
give the exact fraction or decimal or round off final
decimal results to three significant digits.
(Suggestion: When a probability is not a simple
fraction such as 2/3 or 5/9, express it as a decimal
so that the number can be better understood.) All
digits are significant except for the zeros that are
included for proper placement of the decimal point.
Section 4.2-20
Definition
An event is unlikely if its probability is very small,
such as 0.05 or less.
An event has an usually low number of outcomes
of a particular type or an unusually high number
of those outcomes if that number is far from what
we typically expect.
Section 4.2-21
Odds
The actual odds against event A occurring are the ratio
P( A) / P( A), usually expressed in the form of a:b (or “a to b”),
where a and b are integers having no common factors.
The actual odds in favor of event A occurring are the ratio
P( A) / P( A) , which is the reciprocal of the actual odds
against the event. If the odds against A are a:b, then the
odds in favor of A are b:a.
The payoff odds against event A occurring are the ratio of
the net profit (if you win) to the amount bet.
payoff odds against event A = (net profit) : (amount bet)
Section 4.2-22
Example
If you bet \$5 on the number 13 in roulette, your
probability of winning is 1/38 and the payoff odds
are given by the casino at 35:1.
a. Find the actual odds against the outcome of 13.
b. How much net profit would you make if you win
by betting on 13?
Section 4.2-23
Example - continued
a. Find the actual odds against the outcome of 13.
With P(13) = 1/38 and P(not 13) = 37/38, we get:
P  not 13 37 38 37
actual odds against 13 

 , or 37:1.
1
P 13
1
38