Chapter 4 Section 2.ppt

Section 4.2-1
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

Section 4.2-2
Chapter 4
Probability
4-1 Review and Preview
4-2 Basic Concepts of Probability
4-3 Addition Rule
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 4.2-3
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 4.2-4
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 4.2-5
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 4.2-6
Notation for
Probabilities
P - denotes a probability.
A, B, and C - denote specific events.
P(A) - denotes the probability of
event A occurring.

Section 4.2-7
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 4.2-8
Basic Rules for
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 A
P A
n
number of ways can occur
( ) = =
number of different simple events

Section 4.2-9
Basic Rules for
Rule 3: Subjective Probabilities
P(A), the probability of event A, is estimated by
using knowledge of the relevant circumstances.

Section 4.2-10
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 4.2-11
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
three children of the same gender 0.25
8
P  

Section 4.2-12
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 4.2-13
Probability Limits
 The probability of an event that is certain to
occur is 1.
 The probability of an impossible event is 0.
Always express a probability as a fraction or
decimal number between 0 and 1.
 For any event A, the probability of A is
between 0 and 1 inclusive.
That is, .
0 ( ) 1
P A
 

Section 4.2-14
Possible Values
for Probabilities

Section 4.2-15
Complementary Events
The complement of event A, denoted by
, consists of all outcomes in which the
event A does not occur.
A

Section 4.2-16
Example
1010 United States adults were surveyed and 202
of them were smokers.
It follows that:
 
 
202
smoker 0.200
1010
202
not a smoker 1 0.800
1010
 
  
P
P

Section 4.2-17
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 4.2-18
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 4.2-19
The actual odds in favor of event A occurring are the ratio
, 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 actual odds against event A occurring are the ratio
, usually expressed in the form of a:b (or “a to b”),
where a and b are integers having no common factors.
Odds
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)
( )/ ( )
P A P A
( )/ ( )
P A P A

Section 4.2-20
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 4.2-21
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:
, or 37:1.
 
 
37
not 13 37
38
actual odds against 13
1
13 1
38
P
P
  

Section 4.2-22
Example - continued
b. Because the payoff odds against 13 are 35:1,
we have:
$35 profit for each $1 bet. For a $5 bet, there is
$175 net profit. The winning bettor would
collect $175 plus the original $5 bet.

Chapter 4 Section 2.ppt

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