Chapter 3
Probability
3-1
Fundamentals
3-2
Addition Rule
3-3
Multiplication Rule: Basics
3-4
Multiplication Rule: Complements and
Conditional Probability
3-5
Counting Techniques
1
Objectives
 develop sound understanding of
probability values used in subsequent
chapters
 develop basic skills necessary to solve
simple probability problems
2
General Comments
This chapter tends to be the most difficult one
encountered in the course
 Homework note:
Show setup of problem even if using calculator
3
3-1
Fundamentals
Definitions
Experiment – Action being performed
(book uses the word procedure)
Event (E) – A particular observation
within the experiment
Sample space (S) - all possible events
within the experiment
4
Notation
P - denotes a probability
A, B, ... - denote specific events
P (A) -
denotes the probability of
event A occurring
5
Basic Rules for
Computing Probability
Let A equal an Event
P(A) =
number of outcomes favorable to
event “A”
total possible experimental
outcomes(sample space)
6
Probability Limits
 The probability of an impossible event is 0.
 The probability of an event that is certain
to occur is 1.
0  P(A)  1
Impossible
to occur
Certain
to occur
7
Possible Values for Probabilities
1
Certain
Likely
0.5
50-50 Chance
Unlikely
0
Impossible
8
Unlikely Probabilities
Examples:
• Winning the lottery
• Being struck by lightning
• 0.0000035892
• 1 / 727,235
Typically any probability less than 0.05 is
considered unlikely.
9
Example: Roll a die and observe a 4?
Find the
probability.
What is the experiment? Roll a die
What is the event A? Observe a 4
What is the sample space? 1,2,3,4,5,6
Number of outcomes favorable to A is 1.
Number of total outcomes is 6.
What is P(A)?
P(A) = 1 / 6 = 0.167
Similar to #4 on hw
10
Test problem
Example: Toss a coin 3 times and observe exactly 2
heads?
Experiment:
toss a coin 3 times
Event (A):
observe exactly 2 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Note there are 3 outcomes favorable to the event
and 8 total outcomes
Similar to #6 on HW
P(A) = 3 / 8 = 0.375
11
Law of Large Numbers
As a procedure is repeated again and
again, the probability of an event tends
to approach the actual probability.
This is the reason to quit while your ahead
when gambling!
12
Complementary Events
The complement of event A, denoted
by A, consists of all outcomes in
which event A does not occur.
P(A)
P(A)
(read “not A”)
or
c
P(A )
13
Complementary Events
Property of complementary events
P(A) = 1 –
c
P(A )
Example: If the probability of something
occurring is 1/6 what is the probability that it
won’t occur?
14
Example:
A study of randomly selected American
Airlines flights showed that 344 arrived on time and 56
arrived late, What is the probability of a flight arriving
late?
Let
A = late flight
Ac = on time flight
P(Ac) = 344 /(344 + 56) = 344/400 = .86
P(A) = 1 – P(Ac) = 1 - .86 = 0.14
Similar to number 11 & 12 on hw
15
Rounding Off Probabilities
give the exact fraction or decimal
or
round off the final result to three significant
digits
Examples:
•1/3 is exact and could be left as a fraction or
rounded to .333
•0.00038795 would be rounded to 0.000388
16
3-2
Addition Rule
Definitions
Compound Event – Any event combining 2 or
more events
Notation – P(A or B) = P (event A occurs or
event B occurs or they both occur)
General Rule – add the total ways A can occur
and the total way B can occur but don’t double
count
17
Compound Event
• Formal Addition Rule
• P(A or B) = P(A) + P(B) - P(A and B)
where P(A and B) denotes the probability
that A and B both occur at the same time.
• Alternate form
P(AUB) = P(A) + P(B) – P(AB)
18
Definition
Events A and B are mutually
exclusive if they cannot occur
simultaneously.
19
Definition
Not Mutually Exclusive
Mutually Exclusive
P(A or B) = P(A) + P(B) –
P(A and B)
P(A or B) = P(A) + P(B)
Total Area = 1
P(A)
P(B)
Total Area = 1
P(A)
P(B)
P(A and B)
Overlapping Events
Non-overlapping Events
20
Applying the Addition Rule
P(A or B)
Addition Rule
Are
A and B
mutually
exclusive
?
Yes
P(A or B) = P(A) + P(B)
No
P(A or B) = P(A)+ P(B) - P(A and B)
21
Mutually Exclusive
Example: P(A) = 2/7 and P(B) =
3/7 , P(A or B) = 5/7, are A and
B mutually exclusive? Why?
Test question
22
Test Questions
Example: You have an URN with 2 green marbles, 3
red marbles and 4 white marbles
Let A = choose red marble and B = choose white
marble
1. What is the probability of choosing a red marble?
P(A) = 3/9
2. What is the probability of choosing a white
marble? P(B) = 4/9
3. What is the probability of choosing a red or a
white? P(B or A) = 3/9 + 4/9 = 7/9
Why are event A and B mutually
exclusive events?
23
Example:
A card is drawn from a deck of cards.
1. What is the probability that the card is
an ace or jack?
P(ace) + P(jack) = 4/52 + 4/52 = 8/52
2. What is the probability that the card is
an ace or heart?
P(ace) + P(heart) – P(ace of hearts) =
4/52 + 13/52 – 1/52 = 16/52
24
Example: Toss a coin 3 times and observe all
possibilities of the number of heads
Experiment: toss a coin 3 times
Events
(A): observe exactly 0 heads
(B): observe exactly 1 head
(C): observe exactly 2 heads
(D): observe exactly 3 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Find the P(A) + P(B) + P(C) + P(D)
25
Example: Toss a coin 3 times and observe all
possibilities of the number of heads
Event
P(event)
A = 0 heads
1/8
B = 1 head
3/8
C = 2 heads
3/8
D = 3 heads
1/8
Total
1
Probability distribution: Table of all possible events along with
the probability of each event. The sum of all probabilities must
sum to ONE.
Note: Events are mutually exclusive
26
Example: Roll 2 dice and observe the sum
Experiment: roll 2 dice
Event (F): observe sum of 5
Sample Space: 36 elements
One 2 Two 3’s Three 4’s Four 5’s Five 6’s,
One 12 Two 11’s Three 10’s Four 9’s Five 8’s
Six 7’s
Find the P(F)
27
Example: Roll 2 dice and observe the sum
Construct a probability distribution
Event
P(event)
A: Sum = 2
1/36
B: Sum = 3
2/36
And so on…….
Let’s Try #8 From the HW
28
Contingency Table
(Titanic Mortality)
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
• Let A = select a man
• Let B = select a girl
• P(A or B) =
1692
2223
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
* Mutually Exclusive *
+
45
2223
=
1737
2223
= 0.781
29
Contingency Table
(Titanic Mortality)
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
• Let A = select a woman
• Let B = select someone who died.
P(A or B) = (422 + 1517 - 104) / 2223
=
1835 / 2223 = 0.825
* NOT Mutually Exclusive *
Very similar to test problem
30
Contingency Table
(Titanic Mortality)
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
• How could you define a probability distribution for
this data?
31
Complementary Events
P(A) & P(Ac)
P(A) and P(Ac) are mutually
exclusive
P(A) + P(Ac) = 1 (this has to be true)
P(A)

= 1 - P(Ac)
P(Ac) = 1 – P(A)
32
Venn Diagram for the
Complement of Event A
Total Area = 1
P (A)
P (A) = 1 - P (A)
33
3-3
Multiplication Rule
Definitions
Notation: P(A and B)
= P(event A occurs in
a first trial and event B occurs in a second
trial)
Formal Rule
P(A and B) = P(A) • P(B) if independent (with
will define later
replacement)
P(A and B) = P(A) • P(B A) if dependent
(without replacement)
34
Definitions
 Independent Events
Two events A and B are independent if the
occurrence of one does not affect the probability
of the occurrence of the other.
 Dependent Events
If A and B are not independent, they are said to
be dependent.
35
Tree Diagram of Test Answers
T
F
P(T) =
1
2
Ta
Tb
Tc
Td
Te
Fa
Fb
Fc
Fd
Fe
a
b
c
d
e
a
b
c
d
e
P(c) =
1
5
1
P(T and c) = 10
36
P (both correct) = P (T and c)
1 =
1
1
•
10
2
5
Multiplication
Rule
INDEPENDENT EVENTS
37
Independence vs. Dependence
Choose 2 marbles from an URN with 3
red marbles and 3 white marbles
Dependent – choose the 1st marble then
choose the 2nd marble
Independent – choose the 1st marble,
replace it, then choose the 2nd marble
38
Notation for Conditional
Probability
P(B A) represents the probability of
event B occurring after it is assumed that
event A has already occurred (read B A
as “B given A”).
= given
39
Test Questions
Example: You have an URN with 3 red marbles and 4
white marbles
Let A = choose red marble and B = choose white
marble
1. What is the probability of choosing a red marble?
P(A)
2. What is the probability of choosing a white
marble? P(B)
3. If two are chosen find the probability of choosing
a white on the a second trial given a red marble
was chosen 1st. P(B A)
a)
Assume the 1st marble is replaced {independent}
b)
Assume the 1st marble is not replaced {dependent}
40
Example: You have an URN with 3 red marbles and 4
white marbles
Let’s change things a bit….
If two are chosen and we want to find the
probability of choosing a white then
choosing a red marble.
So if we let: A = choose red 1st and B = choose
white 2nd then we need to find P(A and B)
The problem here is that calculating this
probability depends on what happens on
the first draw. We need a rule that helps
us with this.
41
Formal Multiplication Rule
P(A and B) = P(A) • P(B A)
If A and B are independent
events, P(B A) is really the same
as P(B). Will see this in the next
section.
42
Applying the Multiplication Rule
P(A and B)
Multiplication Rule
Are
A and B
independent
?
Yes
P(A and B) = P(A) • P(B)
No
P(A and B) = P(A) • P(B A)
43
Test Questions
Example: You have an URN with 3 red marbles and 4
white marbles
Let A = choose red marble and B = choose white
marble
If two are chosen find the probability of
choosing a red then choosing a white
marble. In other words find P(A and B)
a)
Assume the 1st marble is replaced {independent}
P(A and B) = P(A) • P(B)
b)
Assume the 1st marble is not replaced {dependent}
P(A and B) = P(A) • P(B A)
Use as an example for #6
44
Class Assignment – Part I
You have an URN with 3 red marbles, 7 white marbles,
and 1 green marble
Let
A = choose red
B = choose white
C = choose green
Find the following:
1.
2.
3.
4.
5.
P(Ac), that is find P(not red)
P(A or B)
If two marbles chosen what is the probability that you
choose a white marble 2nd when a red marble was chosen
first. This is, find P(B A)
If two marbles are chosen, find P(A and C) with replacement
If two marbles are chosen, find P(A and C) without
replacement
Very Similar to test question #1
45
Class Assignment – Part II
46
Mutually Exclusive vs.
Independent Events
 Mutually Exclusive Events
P(A or B) = P(A) + P(B)
Independent Events
P(A and B) = P(A) • P(B)
Example:
if P(A) = .3, P(B)=.4, P(A or B)=.7, and
P(A and B) = .12, what can you say about A and B?
Note: Test Question
47
Contingency Table
(Titanic Mortality)
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
• Select two
– Find P(2 women) = 422/2223 x 421/2222
– Find P(2 that died) = 1517/2223 x 1516/2222
• Select one
– Find P(woman and died) = 104/2223
– Find P(Boy and survived) = 29 / 2223
48
3-4 Topics
 Probability of “at least one”
 More on conditional probability
 Test for independence
49
Probability of ‘At Least One’
 ‘At least one’ is equivalent to ‘one or
more’.
 The complement of getting at least one
item of a particular type is that you get no
items of that type.
If P(A) = P(getting at least one), then
P(A) = 1 - P(Ac)
where P(Ac) = P(getting none)
50
Probability of ‘At Least One’
 Find the probability of a getting at least
1 head if you toss a coin 4 times.
P(A) = 1 - P(Ac)
where P(A) is P(no heads)
P(Ac) = (0.5)(0.5)(0.5 )(0.5) = 0.0625
P(A) = 1 - 0.0625 = 0.9375
51
Conditional Probability
P(A and B) = P(A) • P(B|A)
Divide both sides by P(A)
Formal definition for conditional probability
P(B|A) =
P(A and B)
P(A)
52
Testing for Independence
If P(B|A) = P(B)
then the occurrence of A has no effect on the probability
of event B; that is, A and B are independent events.
or
If P(A and B) = P(A) • P(B)
then A and B are independent events. (with replacement)
Example: if A and B are independent, find P(A and B)
if P(A) = 0.3 and P(B) = 0.6 (test question)
53
3-5 Counting
fundamental counting rule
two events (“mn” rule)
multiple events (nr rule)
permutations
factorial rule
different items
not all items different
combinations
54
Fundamental Counting Rule
(‘mn” rule)
If one event can occur m ways and the second
event can occur n ways, the events together
can occur a total of m • n ways.
Example 1: How many ways can your order a meal
with 3 main course choices and 4 deserts? First list
then use rule.
Main Courses
Deserts
Tacos (T)
Ice Cream (IC)
Pasta (P)
Jello (J)
Liver & Onions (LO)
Cake ©
Fruit (F)
55
Fundamental Counting Rule
(nr rule)
If one event that can occur n ways is
repeated r times, the events together
can occur a total of nr ways.
Example 2: How many outcomes are possible when
tossing a coin 3 times? First list then use rule.
Example 3: How many outcomes are possible when
tossing a coin 20 times? Would you care to list all the
outcomes this time?
56
Notation
The factorial symbol ! denotes the product of
decreasing positive whole numbers.
 n! = n (n-1) (n-2) (n-3) • • • • • (3) (2) (1)
Special Definition: 0! = 1
Find the ! key on your calculator
57
Factorial Rule
A collection of n different items can
be arranged in order n! different ways.
Example: How many ways can you order the
letters A, B, C? List first then use rule.
Note: actually a special type of permutation, will define next
58
R
N
Compare the
and
Factorial Rule
Example: How many ways can you order the letters
A, B, C?
a) NR
_______ _________ ________ (with replacement)
b) N!
_______ _________ ________ (without replacement)
59
Permutations Rule
(when items are all different)
 n is the number of available items (without
replacement)
 r is the number of items to be selected
 the number of permutations (or sequences) is
P
n r =
n!
(n - r)!
 Order matters
60
Permutations Rule
(when items are all different)
 Example: Eight men enter a race. In
how many ways can the first 4 positions
be determined?
61
Permutations Rule
( when some items are identical )
If there are n items with n1 alike, n2 alike, .
.
. . nk alike, the number of permutations is
n!
n1! . n2! .. . . . . . . nk!
62
Permutations Rule
( when some items are identical )
Examples:
1. How many ways can you arrange the word
statistics? Or Mississippi?
2. How many ways can you arrange 3 green
marbles and 4 red marbles?
Test Question
63
Permutations Rule
Factorial rule is special case
P
n
n! = n
Can you show this is true?
64
Combinations Rule
 the number of combinations is
n!
nCr = (n - r )! r!
 n different items
 r items to be selected
 different orders of the same items are not
counted (order doesn’t matter)
65
TI-83 Calculator
Calculate n! , nPr, nCr
1. Enter the value for n
2. Press Math
3. Cursor over to Prb
4. Choose 2: nPr or 3: nCr or 4: n! as required
5a. Press Enter for the n! case
5b. Enter the value for “r” for the nPr and nCr
cases
66
Pick five numbers from 1 to 47 and a MEGA number from 1 to 27
Pick five numbers from 1 to 56 and a MEGA number from 1 to 46
Note: game has 2 separate sets of numbers
67
Combinations Rule
 Example: Find the probability of winning
the Pennsylvania Super 6 lotto. Select 6
numbers from 69.
What’s the probably of getting 5 of 6? 4 of
6?, etc. (see lottery handout)
What’s the probability if you have to get all
6 numbers in a specified order?
68
Recall previous example?
Experiment: Toss a coin 3 times
Event (A): Observe 2 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Note there are 3 outcomes favorable to the event
and 8 total outcomes
P(A) = 3 / 8 = 0.375
Test problem
69
Let’s take a different approach
Experiment: Toss a coin 3 times
Event (A): Observe 2 heads
There are 23 possible outcomes
and 3C2 ways to get 2 heads
P(A) = 3C2 / 23 = 3 / 8 = 0.375
Test problem
70
Example:
Experiment: Toss a coin 6 times
Event (A): Observe 4 heads
There are 26 possible outcomes
and 6C4 ways to get 4 heads
P(A) = 6C4 / 26 = 15 / 64 =
0.234
Test problem
71