Download br7ch04 - Web4students

Document related concepts

Probability wikipedia , lookup

Transcript
Understandable Statistics
Seventh Edition
By Brase and Brase
Prepared by: Lynn Smith
Gloucester County College
Chapter Four
Elementary Probability Theory
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
1
Probability
• Probability is a numerical measurement
of likelihood of an event.
• The probability of any event is a number
between zero and one.
• Events with probability close to one are
more likely to occur.
• If an event has probability equal to one,
the event is certain to occur.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
2
Probability Notation
If A represents an event,
P(A)
represents the probability of A.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
3
Three methods to find
probabilities:
• Intuition
• Relative frequency
• Equally likely outcomes
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
4
Intuition method
based upon our level of confidence
in the result
Example: I am 95% sure that I
will attend the party.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
5
Probability
as Relative Frequency
Probability of an event =
the fraction of the time that the event
occurred in the past =
f
n
where f = frequency of an event
n = sample size
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
6
Example of Probability
as Relative Frequency
If you note that 57 of the last 100 applicants
for a job have been female, the
probability that the next applicant is
female would be:
57
100
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
7
Law of Large Numbers
In the long run,
as the sample size increases and increases,
the relative frequencies of outcomes
get closer and closer to
the theoretical (or actual) probability value.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
8
Equally likely outcomes
No one result is expected to occur
more frequently than any other.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
9
Probability of an event when
outcomes are equally likely =
number of outcomes favorable to event
total number of outcomes
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
10
Example of Equally Likely
Outcome Method
When rolling a die, the probability of
getting a number less than three =
2 1

6 3
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
11
Statistical Experiment
activity that results in a definite
outcome
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
12
Sample Space
set of all possible outcomes of an
experiment
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
13
Sample Space for the rolling of
an ordinary die:
1, 2, 3, 4, 5, 6
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
14
For the experiment of
rolling an ordinary die:
• P(even number) =
3 = 1
6
2
• P(result less than four) = 3 = 1
6
2
5
• P(not getting a two) =
6
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
15
Complement of Event A
the event not A
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
16
Probability of a Complement
P(not A) = 1 – P(A)
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
17
Probability of a Complement
If the probability that it will snow
today is 30%,
P(It will not snow) = 1 – P(snow) =
1 – 0.30 = 0.70
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
18
Probabilities of an Event and
its Complement
• Denote the probability of an event by the
letter p.
• Denote the probability of the complement
of the event by the letter q.
• p + q must equal 1
• q=1-p
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
19
Probability
Related to Statistics
• Probability makes statements about what
will occur when samples are drawn from
a known population.
• Statistics describes how samples are to be
obtained and how inferences are to be
made about unknown populations.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
20
Independent Events
The occurrence (or non-occurrence)
of one event does not change the
probability that the other event will
occur.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
21
If events A and B are
independent,
P(A and B) = P(A) P(B)
•
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
22
Conditional Probability
• If events are dependent, the occurrence of
one event changes the probability of the
other.
• The notation P(A|B) is read “the
probability of A, given B.”
• P(A, given B) equals the probability that
event A occurs, assuming that B has
already occurred.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
23
For Dependent Events:
• P(A and B) = P(A) P(B, given A)
•
• P(A and B) = P(B) P(A, given B)
•
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
24
The Multiplication Rules:
• For independent events:
P(A and B) = P(A) P(B)
•
• For dependent events:
P(A and B) = P(A) P(B, given A)
•
P(A and B) = P(B) P(A, given B)
•
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
25
For independent events:
P(A and B) = P(A) P(B)
•
When choosing two cards from two
separate decks of cards, find the
probability of getting two fives.
P(two fives) =
P(5 from first deck and 5 from second) =
1 1
1
 
13 13 169
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
26
For dependent events:
P(A and B) = P(A) P(B, given A)
•
When choosing two cards from a deck
without replacement, find the probability
of getting two fives.
P(two fives) =
P(5 on first draw and 5 on second) =
4 3
12
1
 

52 51 2652 221
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
27
“And” versus “or”
• And means both events occur together.
• Or means that at least one of the events
occur.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
28
For any events A and B,
P(A or B) =
P(A) + P(B) – P(A and B)
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
29
When choosing a card from an
ordinary deck, the probability
of getting a five or a red card:
P(5 ) + P(red) – P(5 and red) =
4 26 2 28 7




52 52 52 52 13
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
30
When choosing a card from an
ordinary deck, the probability
of getting a five or a six:
P(5 ) + P(6) – P(5 and 6) =
4
4
0
8
2




52 52 52 52 13
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
31
For any mutually exclusive
events A and B,
P(A or B) = P(A) + P(B)
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
32
When rolling an ordinary die:
P(4 or 6) =
1 1 2 1
  
6 6 6 3
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
33
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
P(male and college grad) = ?
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
34
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
54
P(male and college grad) =
187
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
35
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
P(male or college grad) = ?
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
36
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
P(male or college grad) =
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
147
187
37
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
P(male, given college grad) = ?
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
38
Survey results:
Education:
Males
Females
Row totals
College
Graduates
54
62
116
Not College 31
Graduates
40
71
Column
totals
102
187(Grand total)
85
P(male, given college grad) = 54
116
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
39
Counting Techniques
• Tree Diagram
• Multiplication Rule
• Permutations
• Combinations
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
40
Tree Diagram
a method of listing outcomes of an
experiment consisting of a series of
activities
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
41
Tree diagram for the
experiment of tossing two coins
H
H
start
T
H
T
T
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
42
Find the number of paths
without constructing the tree
diagram:
Experiment of rolling two dice, one
after the other and observing any
of the six possible outcomes each
time .
Number of paths = 6 x 6 = 36
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
43
Multiplication of Choices
If there are
n possible outcomes for event E1
and m possible outcomes for event E2,
then there are
n x m or nm possible outcomes
for the series of events E1 followed by E2.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
44
Area Code Example
Until a few years ago a three-digit area code was
designed as follows.
The first could be any digit from 2 through 9.
The second digit could be only a 0 or 1.
The last could be any digit.
How many different such area codes were possible?
8 2 10 = 160

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .

45
Ordered Arrangements
In how many different ways could four
items be arranged in order from first to
last?
4  3  2  1 = 24
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
46
Factorial Notation
• n! is read "n factorial"
• n! is applied only when n is a whole
number.
• n! is a product of n with each positive
counting number less than n
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
47
Calculating Factorials
5! = 5 • 4 • 3 • 2 • 1 =
3! = 3 • 2 • 1 =
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
120
6
48
Definitions
1! = 1
0! = 1
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
49
Complete the Factorials:
4! = 24
10! = 3,628,800
6! = 720
15! = 1.3077 x 1012
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
50
Permutations
A permutation is an arrangement in a
particular order of a group of items.
There are to be no repetitions of items
within a permutation.)
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
51
Listing Permutations
How many different permutations of the
letters a, b, c are possible?
Solution: There are six different
permutations:
abc, acb, bac, bca, cab, cba.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
52
Listing Permutations
How many different two-letter
permutations of the letters a, b, c, d are
possible?
Solution: There are twelve different
permutations:
ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
53
Permutation Formula
The number of ways to arrange in
order n distinct objects, taking them
r at a time, is:
Pn , r
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
n!

n  r  !
54
Another notation for
permutations:
P
n r
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
55
Find P7, 3
7!
7! 5040
P7 , 3 
 
 210
(7  3)! 4!
24
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
56
Applying the Permutation
Formula
6
P3, 3 = _______
30
P6, 2
= __________
P15, 2
= _______
12
P4, 2 = _______
336
P8, 3 = _______
210
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
57
Application of Permutations
A teacher has chosen eight possible questions
for an upcoming quiz. In how many different
ways can five of these questions be chosen and
arranged in order from #1 to #5?
Solution: P8,5 =
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
8!
3!
= 8• 7 • 6 • 5 • 4 = 6720
58
Combinations
A combination is a grouping
in no particular order
of items.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
59
Combination Formula
The number of combinations of n objects
taken r at a time is:
Cn , r
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
n!

(n  r ) ! r !
60
Other notations for
combinations:
C
n
r
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
n
or  
r
 
61
Find C9, 3
C9 , 3
9!
9! 362880



 84
3!(9  3)! 3!6! 6(720)
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
62
Applying the Combination
Formula
10
C5, 3 = ______
35
C7, 3 = ________
1
C3, 3 = ______
105
C15, 2 = ________
15
C6, 2 = ______
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
63
Application of Combinations
A teacher has chosen eight possible questions
for an upcoming quiz. In how many different
ways can five of these questions be chosen if
order makes no difference?
Solution: C8,5
8!
=
= 56
5!3 !
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved .
64