STAT1010 – Intro to probability part 3
6.2 Introduction to Probability
part 2
!  From Tables to Probability
"  Computing relative frequency probabilities from a
table.
!  Probability
Distributions
1
Summary Table
!  The
table below provides information on the
hair color and eye color of 592 statistics
students (The American Statistician, 1974).
Row
Eye Color
Brown
Blue
Hazel
Green
Total
Black
68
20
15
5
108
Hair
Brown
119
84
54
29
286
Color
Red
26
17
14
14
71
Blond
7
94
10
16
127
592
2
Row
Eye Color
Brown
Blue
Hazel
Green
Total
Black
68
20
15
5
108
Hair
Brown
119
84
54
29
286
Color
Red
26
17
14
14
71
Blond
7
94
10
16
127
592
!  If
a student is drawn at random from this
population…
" What
is the probability that they are blond?
!  ANS:
" What
127/592 = 0.2145
is the probability that they are NOT blond?
!  ANS:
!  ANS:
P(NOT blond) = 1-P(blond) =1-127/592 = 0.7855
(71 +286 + 108)/592 = 465/592 = 0.7855
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STAT1010 – Intro to probability part 3
Row
Eye Color
Brown
Blue
Hazel
Green
Total
Black
68
20
15
5
108
Hair
Brown
119
84
54
29
286
Color
Red
26
17
14
14
71
Blond
7
94
10
16
127
592
!  If
a student is drawn at random from this
population…
" What
is the probability that they are blond
and have brown eyes?
!  ANS:
7/592 = 0.0118
" What
is the probability that they have red
hair and have green eyes?
!  ANS:
14/592 = 0.0236
4
Row
Eye Color
Brown
Blue
Hazel
Green
Total
Black
68
20
15
5
108
Hair
Brown
119
84
54
29
286
Color
Red
26
17
14
14
71
Blond
7
94
10
16
127
592
!  Based
on this table…
" What
is the probability that a blond student
has brown eyes?
!  ANS:
7/127 = 0.0551
We’ve subsetted to a smaller
population from which we are
drawing a person
" What
is the probability that a brown-haired
student has hazel eyes?
!  ANS:
54/286 = 0.1888
5
Relative Frequency Method
!  Step
1. Repeat or observe a process
many times and count the number of times
the event of interest, A, occurs.
!  Step
! 
2. Estimate P(A) by
P(A) =
number of times A occurred
total number of observations
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2
STAT1010 – Intro to probability part 3
Example: 500-year flood
!  Geological
records indicate that a river has
crested above a particular high flood level four
times in the past 2,000 years. What is the
relative frequency probability that the river
will crest above the high flood level next year?
number of years with flood
total number of years
=
4 = 1
2000 500
The probability of having a flood of this magnitude in
any given year is 1/500, or 0.002.
7
Probability Distributions
!  A probability
distribution shows the possible
values that could occur, and the probability
for each occurring.
The probability
distribution for the
results of tossing
two coins.
8
Probability Distributions
!  A probability
distribution shows the possible
values that could occur, and the probability
for each occurring.
The possible values can be
seen on the horizontal axis.
9
3
STAT1010 – Intro to probability part 3
Probability Distributions
!  A probability
distribution shows the possible
values that could occur, and the probability
for each occurring.
The probability of each value is
shown by the height of the bar.
10
Example: Roll two 6-sided dice
!  Find
the probability distribution for the ‘sum
of two dice’.
Most likely sum
is a sum of 7.
0.18
0.16
0.14
0.12
Probability
0.10
0.08
Only one way
to get sum of
12 (double 6’s).
0.06
0.04
0.02
2
3
4
5
6
7
Sum of two dice
8
9
10
11
P(sum of 12) =
1/36 = 0.02778
12
11
Example: Roll two 6-sided dice
!  Find
the probability distribution for the ‘sum
of two dice’.
0.18
0.16
0.14
0.12
0.10
Probability
Recall,
P(sum of 11) =
2/36 = 0.0556
And that is the
height of the bar
above the number
“11”.
0.08
0.06
0.04
0.02
2
3
4
5
6
7
Sum of two dice
8
9
10
11
12
12
4
STAT1010 – Intro to probability part 3
Example: Roll one 6-sided dice
the probability distribution for ‘the
number that turns up’.
0.6
0.4
0.0
Each of 6 values is
equally likely, at 1/6
= 0.1667
0.2
probability
0.8
1.0
!  Find
1
2
3
4
5
6
value on die
This is a uniform distribution.
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Probability Distributions
!  How
1) 
2) 
3) 
to make a probability distribution.
List all possible outcomes. Use a table or
figure it is helpful.
Identify outcomes that represent the same
event. Find the probability of each event.
Make a table in which one column lists
each event and another column lists each
probability. The sum of all the probabilities
must be 1.
14
Example: Making a probability
Probability Distribution
!  Random
experiment: Toss Three Coins
!  Make a probability distribution for the
number of heads that occurs.
1)  List the possible outcomes (equally likely):
HHH,HHT,HTH,THH,HTT,THT,TTH,TTT
2)  Identify outcomes that represent the same
event. There are four possible events:
0, 1, 2, or 3 heads.
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5
STAT1010 – Intro to probability part 3
Example: Making a probability
Probability Distribution
!  Random
experiment: Toss Three Coins
a probability distribution for the
number of heads that occurs.
!  Make
3) Make
a table
0: TTT (1 outcome)
1: HTT,THT,TTH (3 outcomes)
2: HHT,HTH,THH (3 outcomes)
3: HHH (1 outcome)
Numberofheads Probability
0
1/8
1
3/8
2
3/8
3
1/8
This column sums to 1.
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