Transcript lecture 1 probability

TR 555 Statistics “Refresher”
Lecture 1: Probability Concepts
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References:
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Penn State University, Dept. of Statistics
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Statistics: Making Sense of Data (MIT)
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1
Statistical Education Resource Kit
a collection of resources used by faculty in Penn State's
Department of Statistics in teaching introductory statistics
courses.
Page maintained by Laura J. Simon, Sept. 2003
William Stout, John Marden and Kenneth Travers
http://www.introductorystatistics.com/ Sept. 2003
Tom Maze, stat course prepared for KDOT, 2003
Outline
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2
Overview of statistics
Types of data
Describing data numerically and graphically
Probability and random variables
Probability and Statistics

Probably is the likelihood of an event occurring relative to all
other events
–
Example:
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If a coin is flipped, what is the probability of getting a heads
–

Given that the last flip was a heads what is the probability that the next will
be heads
–
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0.5
Statistics is the measurement and modeling of random variables
–
Example:
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3
0.5
If our state averages 200 fatal crashes per year, what is the probability of
having one crash today. Poisson distribution – k = average per time
period. 200/365 = 0.55
–
P(1 = x) = ((kt)x/x!)e-kt=(0.55*1)1/1!)e-0.55(1)= 0.32
Data Collection
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Designing experiments
–
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Observational studies
–
4
Does aspirin help reduce the risk of heart
attacks?
Polls - Clinton’s approval rating
Variable Types

Deterministic
–
–
–
Assume away variation and randomness
Known with certainty
One to one mapping of independent variable to
dependent variable
Relationship
Y1
X1
5
Variable Types Continued
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Random or Stochastic
–
–
Recognized uncertainty of an event
One to one distribution mapping of independent
variable to dependent variable
Less
Likely
6
Most Likely
Less Likely
Probability that it could be any of these values
Population
The set of data (numerical or otherwise)
corresponding to the entire collection of units
about which information is sought
7
Sample
A subset of the population data that are
actually collected in the course of a study.
8
WHO CARES?
In most studies, it is difficult to
obtain information from the
entire population. We rely on
samples to make estimates or
inferences related to the
population.
9
Organization and Description of
Data
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Qualitative vs. Quantitative data
Discrete vs. Continuous Data
Graphical Displays
Measures of Center
Measures of Variation
Qualitative (Categorical) Data
The raw (unsummarized) data are
merely labels or categories
Quantitative (Numerical) Data
The raw (unsummarized) data are
numerical
11
Qualitative Data Examples
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12
Class Standing (Fr, So, Ju, Sr)
Section # (1,2,3,4,5,6)
Automobile Make (Ford, Chevrolet, Nissan)
Questionnaire response (disagree, neutral,
agree)
Quantitative Data Examples
(measures)
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13
Voltage
Height
Weight
SAT Score
Number of students arriving late for class
Time to complete a task
Discrete Data
Only certain values are possible (there are
gaps between the possible values)
Continuous Data
Theoretically, any value within an interval is
possible with a fine enough measuring
device
14
Discrete Data Examples
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Number of students late for class
Number of crimes reported to SC police
Number of times the word number is used
(generally, discrete data are counts)
15
Discrete Variable Model
Poisson Distribution
Probability of # of Fatals per one day
0.7
(0.55*t)x/x!)e-0.55(t)
0.6
Probabilty
0.5
0.4
0.3
0.2
0.1
0
0
16
2
1
4
3
6
5
8
7
10
9
# of Fatal Crashes
12
11
14
13
15
Continuous Data Examples
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Voltage
Height
Weight
Time to complete a homework assignment
Continuous Variable Model
Exponential Distribution
Fatality Probability Density Function
0.6
Probability
0.5
0.4
0.3
Probability of first
Fatal at time t
= ke-tk
0.2
0.1
0
18
0
0.8
1.6
2.4
3.2
4
Time till the first fatal accident
4.7
5.5
Continuous Probability Function
Cummulative Probability till first fatal
Cummulative Probability
1.2
1
0.8
0.6
Cumulative Probability of
Time Till First Fatal t = 1 - e-tk
0.4
0.2
0
0
19
0.8
1.6
2.4
3.2
Days
4
4.7
5.5
Nominal Data
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A type of categorical data in which objects fall
into unordered categories, for example:
–
Hair color
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–
Race
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–
Caucasian, African-American, Asian, etc.
Smoking status
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20
blonde, brown, red, black, etc.
smoker, non-smoker
Ordinal Data
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A type of categorical data in which order is
important. For example …
–
Class
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Degree of illness
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–
none, mild, moderate, severe, …, going, going, gone
Opinion of students about riots
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fresh, sophomore, junior, senior, super senior
ticked off, neutral, happy
Binary Data
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A type of categorical data in which there are only
two categories.
Binary data can either be nominal or ordinal, for
example …
–
Smoking status
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–
Attendance
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–
present, absent
Class
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smoker, non-smoker
lower classman, upper classman
Interval and Ratio Data
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Interval
–
–
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Ratio
–
–
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Interval is important, but no meaningful zero
e.g, temperature in farenheit
has a meaningful zero value
e.g., temperature in Kelvin, crash rate
Who Cares?
The type(s) of data collected in
a study determine the type of
statistical analysis used.
24
Proportions

Categorical data are commonly summarized
using “percentages” (or “proportions”).
–
–
25
11% of students have a tattoo
2%, 33%, 39%, and 26% of the students in class
are, respectively, freshmen, sophomores, juniors,
and seniors
Averages
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Measurement data are typically summarized
using “averages” (or “means”).
–
–
–
26
Average number of siblings Fall 1998 Stat 250
students have is 1.9.
Average weight of male Fall 1998 Stat 250
students is 173 pounds.
Average weight of female Fall 1998 Stat 250
students is 138 pounds.
Descriptive statistics
Describing data with
numbers:
measures of location
27
Mean
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Another name for average.
If describing a population, denoted as , the
greek letter “mu”.
_
If describing a sample, denoted as x, called
“x-bar”.
Appropriate for describing measurement
data.
Seriously affected by unusual values called
“outliers”.
Calculating Sample Mean
Formula:
Xi

X n
That is, add up all of the data points and divide by the
number of data points.
Data (# of classes skipped): 2
8
3
4
Sample Mean = (2+8+3+4+1)/5 = 3.6
29
Do not round! Mean need not be a whole number.
1
Population Mean
The mean of a random variable X is called the
population mean and is denoted 
It is also called the expected value of X or the
expectation of X and is denoted E(X).
  E ( X )   xi f  xi 
30
Median
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Another name for 50th percentile.
Appropriate for describing measurement
data.
“Robust to outliers,” that is, not affected
much by unusual values.
Calculating Sample Median
Order data from smallest to largest.
If odd number of data points, the median is
the middle value.
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Data (# of classes skipped): 2
8
Ordered Data: 1
8
2
3
4
Median
3
4
1
Calculating Sample Median
Order data from smallest to largest.
If even number of data points, the median is
the average of the two middle values.
Data (# of classes skipped): 2 8 3 4 1 8
Ordered Data: 1
33
2
3
4
8
8
Median = (3+4)/2 = 3.5
Mode
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The value that occurs most frequently.
One data set can have many modes.
Appropriate for all types of data, but most
useful for categorical data or discrete data
with only a few number of possible values.
Most appropriate
measure of location
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Depends on whether or not data are
“symmetric” or “skewed”.
Depends on whether or not data have one
(“unimodal”) or more (“multimodal”) modes.
Symmetric and Unimodal
Percent
20
10
0
2
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
GPAs
36
3
GPA
4
Symmetric and Bimodal
37
Skewed Right
Number of Music CDs of Spring 1998 Stat 250 Students
20
10
0
0
0
100
200
300
Number of Music CDs
38
400
100
200
Number of CDs
300
400
Skewed Left
30
Percent
20
10
0
50
55
60
65
70
75
80
grades
39
85
90
95 100
Choosing Appropriate
Measure of Location
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If data are symmetric, the mean, median, and
mode will be approximately the same.
If data are multimodal, report the mean,
median and/or mode for each subgroup.
If data are skewed, report the median.
Descriptive statistics
Describing data with
numbers: measures of
variability
41
Range
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The difference between
largest and smallest
data point.
Highly affected by
outliers.
Best for symmetric data
with no outliers.
Frequency

GPAs of Spring 1998 Stat 250 Students
20
10
0
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
GPA
42
Interquartile range
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The difference between the
“third quartile” (75th
percentile) and the “first
quartile” (25th percentile).
So, the “middle-half” of the
values.
IQR = Q3-Q1
Robust to outliers or
extreme observations.
Works well for skewed data.
Frequency
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GPAs of Spring 1998 Stat 250 Students
20
10
0
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
GPA
Variance
2
(x

x
)
s2  
n 1
44
1. Find difference between
each data point and mean.
2. Square the differences, and
add them up.
3. Divide by one less than the
number of data points.
Variance
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If measuring variance of population, denoted
by 2 (“sigma-squared”).
If measuring variance of sample, denoted by
s2 (“s-squared”).
Measures average squared deviation of data
points from their mean.
Highly affected by outliers. Best for
symmetric data.
Problem is units are squared.
Population Variance
The variance of a random variable X is called
2
the population variance and is denoted 
    xi    f  xi 
2
46
2
Standard deviation
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Sample standard deviation is square root of
sample variance, and so is denoted by s.
Units are the original units.
Measures average deviation of data points
from their mean.
Also, highly affected by outliers.
Population Standard Deviation
The population standard deviation is the square
root of the population variance and is
denoted 
  
2
48
 x    f x 
2
i
i
What is the variance
or standard deviation?
Fastest Ever Driving Speed
226 Stat 100 Students, Fall '98
100
Men
126
Women
70
49
80
90
100 110 120 130 140 150 160
Speed (MPH)
Variance or standard deviation
Sex
N
female 126
male
100
female
male
Mean
91.23
06.79
Minimum
65.00
75.00
Median
90.00
110.00
Maximum
120.00
162.00
TrMean
90.83
105.62
StDev SE Mean
11.32
1.01
17.39
1.74
Q1
85.00
95.00
Q3
98.25
118.75
Females: s = 11.32 mph and s2 = 11.322 = 128.1 mph2
Males: s = 17.39 mph and s2 = 17.392 = 302.5 mph2
50
Coefficient of Variation (COV) – not
covariance!
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Ratio of sample standard deviation to sample
mean multiplied by 100.
Measures relative variability, that is,
variability relative to the magnitude of the
data.
Unitless, so good for comparing variation
between two groups.
Coefficient of variation (MPH)
Sex
N
Mean
female 126 91.23
male
100 106.79
female
male
Minimum
65.00
75.00
Median
90.00
110.00
Maximum
120.00
162.00
TrMean
90.83
105.62
Q1
85.00
95.00
Females: CV = (11.32/91.23) x 100 = 12.4
Males: CV = (17.39/106.79) x 100 = 16.3
52
StDev SE Mean
11.32
1.01
17.39
1.74
Q3
98.25
118.75
Choosing Appropriate
Measure of Variability
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53
If data are symmetric, with no serious
outliers, use range and standard deviation.
If data are skewed, and/or have serious
outliers, use IQR.
If comparing variation across two data sets,
use coefficient of variation.
Descriptive Statistics
Summarizing data using
graphs
54
Which graph to use?
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Depends on type of data
Depends on what you want to illustrate
Depends on available statistical software
Birth Order of Spring 1998 Stat 250 Students
40
30
Percent
Bar Chart
20
10
Middle
Oldest
Only
Youngest
Birth Order
n=92 students
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Summarizes categorical data.
Horizontal axis represents categories, while vertical
axis represents either counts (“frequencies”) or
percentages (“relative frequencies”).
Used to illustrate the differences in percentages (or
counts) between categories.
Age of Spring 1998 Stat 250 Students
Frequency (Count)
50
Histogram
40
30
20
10
0
18
19
20
21
22
23
24
25
26
27
Age (in years)
n=92 students
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57
Divide measurement up into equal-sized categories.
Determine number (or percentage) of
measurements falling into each category.
Draw a bar for each category so bars’ heights
represent number (or percent) falling into the
categories.
Label and title appropriately.
Number of ranges (see Tufte)
Use common sense in determining
number of categories to use.
(Trial-and-error works fine, too.)
Age of Spring 1998 Stat 250 Students
GPAs of Spring 1998 Stat 250 Students
7
50
6
Frequency (Count)
60
40
30
20
10
4
3
2
1
0
0
58
5
18
23
2
28
Age (in years)
n=92 students
3
GPA
n=92 students
4
Dot Plot
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Summarizes
measurement
data.
Horizontal axis
represents
measurement
scale.
Plot one dot for
each data point.
Fastest Ever Driving Speed
226 Stat 100 Students, Fall '98
100
Men
126
Women
70
80
90
100 110 120 130 140 150 160
Speed
Stem-and-Leaf Plot
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60
Summarizes
measurement data.
Each data point is
broken down into a
“stem” and a “leaf.”
First, “stems” are
aligned in a column.
Then, “leaves” are
attached to the stems.
Boxplot
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.
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smallest observation = 3.20
Q1 = 43.645
Q2 (median) = 60.345
Q3 = 84.96
largest observation = 124.27
. .
.
.
0 10 20 30 40 50 60 70 80 90 100 110 120 130
Amount of sleep in past 24 hours
of Spring 1998 Stat 250 Students
10
9
Hours of sleep
Box Plot
8
7
6
5
4
3
2
1
0

“Whiskers” are drawn to the most extreme data points
that are not more than 1.5 times the length of the box
beyond either quartile.
–

“Outliers,” or extreme observations, are denoted by
asterisks.
–
62

Whiskers are useful for identifying outliers.
Generally, data points falling beyond the whiskers are
considered outliers.
Useful for comparing two distributions
Using Box Plots to Compare
Fastest Ever Driving Speed
226 Stat 100 Students, Fall 1998
160
110
60
63
female
male
Gender
Scatter Plots
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64
Summarizes the
relationship between
two measurement
variables.
Horizontal axis
represents one variable
and vertical axis
represents second
variable.
Plot one point for each
pair of measurements.
Right foot (in cm)

Foot sizes of Spring 1998 Stat 250 students
31
30
29
28
27
26
25
24
23
22
22
23
24
25
26
27
28
Left foot (in cm)
n=88 students
29
30
31
No relationship
Lengths of left forearms and head circumferences
of Spring 1998 Stat 250 Students
32
31
30
29
28
27
26
25
24
23
22
52
65
57
Head circumference (in cm)
n=89 students
62
Closing comments
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Many possible types of graphs.
Use common sense in reading graphs.
When creating graphs, don’t summarize your
data too much or too little.
When creating graphs, label everything for
others. Remember you are trying to
communicate something to others!
Probability
You’ll probably like it!
67
Before we begin …

What is the probability that 2 or more people
share the same birthday if …
–
–
–
–
68
5 people are in the sample?
23 people?
50 people?
This class?
Probability Properties
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69
The probability of an event “A” (the
proportion of times the event is expected to
occur in repeated experiments), is denoted
P(A).
All probabilities are between 0 and 1.
(i.e. 0 < P(A) < 1)
The sum of the probabilities of all possible
outcomes must be 1.
Probability Basics
 Given
that a crash has occurred, what is the
probability that it is a fatal crash?
events – Fatal, injury, and property
damage only
– Possible
 Fatal
 Injury
 PDO
 Total
70
Crashes
37,000
P(F) = 0.58%
2,026,000 P(I) = 32.16%
4,226,000 P(D) = 67.08%
6,300,000
Complement
The complement of an event A, denoted by A, is the set
of outcomes that are not in A
A means A does not occur
P(A) = 1 - P(A)
Some texts use Ac to denote the complement of A
71
Union
The union of two events A and B, denoted by
A U B, is the set of outcomes that are in A,
or B, or both
If A U B occurs, then either A or B or both
occur
72
Intersection
The intersection of two events A and B,
denoted by AB, is the set of outcomes that
are in both A and B.
If AB occurs, then both A and B occur
73
Combinations of Events
All Fatal Crashes (37,795)
Union of fatal speed related and run-off the road crashes
21,052
Speed Related
Crashes
13,357
74
Single Vehicle
Crash
Intersection of Fatal and Run-off the Road Crashes
Addition Law
P(A U B) = P(A) + P(B) - P(AB)
(The probability of the union of A and B is the
probability of A plus the probability of B
minus the probability of the intersection of A
and B)
75
Mutually Exclusive Events
Two events are mutually exclusive if
their intersection is empty.
Two events, A and B, are mutually
exclusive if and only if P(AB) = 0
P(A U B) = P(A) + P(B)
76
Conditional Probability
The probability of event A occurring, given that
event B has occurred, is called the conditional
probability of event A given event B, denoted
P(A|B)
77
Multiplication Rule
 General
form P(A/B) = P(A,B)/P(B)
 e.g., what is the probability of a single vehicle
accident given that it was speed related?
78
Conditional Probability Example





Total fatal crashes - 37,795
Total speed related crashes – 13,357
Total single vehicle crashes – 21,052
Total single vehicle, speed related crashes - 8,600
If the crash was speed related, what is the probability that it was a
single vehicle crash?
–

If the crash was speed related, what is the probability that it was not a
single vehicle crash?
–
79
P(sv/sp) = 8600/13357 = 64.38%
P(sv/sp) = 1 – 0.6438 = 35.62%
All Fatal
Crashes
37,795
Speed Related SR+SV Single Vehicle
Crashes
Crashes
8,600
21,052
13,357
Conditional Probability Example (Cont)

Probability that a fatal crash was speed related = P(sp)
–

Probability that a fatal crash was a single vehicle = P(sv)
–

13,357/ 37,795 = 35.34%
21,052/37,795 = 55.70%
Probability that a fatal crash is both speeding related and a
single vehicle = P(sv,sp)
–
8,600/37,795 = 22.74%
All Fatal
Crashes
37,795
Speed Related SR+SV Single Vehicle
Crashes
Crashes
8,600
21,052
13,357
80
Bayes’ Theorem
 P(A/B)P(B)
= P(B/A)P(A)
 P(B/A) = P(A/B)P(B)/P(A)
 P(sv) = 55.70%
 P(sp) = 35.34%
 P(sv/sp) = 64.38%
 P(sp/sv) = ?
 P(sp/sv) =
((0.6438)*(0.3534))/0.5570
= 0.3854
81
All Fatal
Crashes
37,795
Speed Related SR+SV Single Vehicle
Crashes
Crashes
8,600
21,052
13,357
Bayes’ Theorem Problem
 Given
There were 11,696 off-road fixed object fatal crashes involving
a single vehicle
– There were 13,357 fatal crashes involving a speeding vehicle
– There were 8,600 fatal crashes involving speeding and single
vehicles
– There were 5,400 fatal crashes involving single vehicles,
speeding, and off-road fixed object crashes
– The total number of fatal crashes is 37,795
– Given that a crash is speeding related, what is the probability
that it will be an off-road single vehicle crash
–
82
Bayes’ Problem Answer
 What
we need to know P(or,sv/sp)
 What we know
– P(or,sv)
= 30.95%
– P(sp) = 35.34%
– P(sv,sp) = 55.70%
– P(sv,sp) = 22.75%
– P(sp,or,sv) = 14.29%
– P(or,sv/sv) = 55.56%
83
Answer Continued
 Multiplication
Rule
– P(sp/or,sv)P(or,sv)
= P(sp,or,sv)
– P(sp/or,sv) = P(sp,or,sv)/P(or,sv)
– 46.17% =0.1429/0.3095
 Bayes’
Theorem
– P(or,sv/sp)=
(P(sp/or,sv)*P(or,sv))/P(sp)
– 40.43% = (0.4617*0.3095)/0.3534
84
Independence
Two events A and B are independent if
P(A|B) = P(A)
or
P(B|A) = P(B)
or
P(AB) = P(A)P(B)
85
Probability Concepts
Randomness
Independence
86
Thought Question 1
What does it mean to say that a deck of
cards is “randomly” shuffled?

Every ordering of the cards is equally likely
There are 8 followed by 67 zeros possible orderings
of a 52 card deck

87
Every card has the same probability to end up
in any specified location
The question continued
A 52 card deck is randomly shuffled
How often will the tenth card down from
the top be a Club?


88
1/4 of the time
Every card has the same chance to end up
10th. There are 13 clubs and 13 / 52 = 1/4
Law of Large Numbers
Relative frequency of an event gets closer
to true probability as number of trials gets
larger
89
Probability values
Probabilities are between 0 and 1
Total probabilities of all possible
outcomes = 1
Probability = 1

means an event always happens
Probability = 0

90
means an event never happens
Does a prior event matter?
A fair coin is flipped four times.
First three flips are heads
What’s the probability that the fourth flip
is heads?
1/2 assuming flips are independent

91
Results of first three flips don’t matter
Independence
The chance that B happens is not affected
by whether A had happened.
92
Does prior event matter?
Ten card drawn without replacement from
52 card deck.
2 Aces are among these 10 cards
What’s the probability the tenth card is an
Ace?
2/42 = 1/21

93
After ten draws, 42 cards remain, 2 of them are
Aces
Dependence
The chance that B happens is affected by
whether A has happened.
94
Sequence of Events
You guess at five True False questions.
What’s the probability you get them right?
95
Five right in five guesses
For each question, Pr(correct) = 1/2
Multiply probabilities

96
(1/2) x (1/2) x (1/2) x (1/2) x (1/2) = 1/32 = 0.031
Card Example
Two cards are taken from normal 52 card
deck.
What’s the probability that both are
Hearts?
Note - there’s dependence between the
two cards
Answer = (13/52) x (12/51) = 1/17 = 0.059
97
The Birthday Problem

98
What is the probability that at least two
people in this class share the same birthday?
Assumptions


99
Only 365 days each year.
Birthdays are evenly distributed throughout
the year, so that each day of the year has an
equal chance of being someone’s birthday.
Take group of 5 people….
Let A = event no one in group shares same birthday.
Then AC = event at least 2 people share same birthday.
P(A) = 365/365 × 364/365 × 363/365 × 362/365 × 361/365
= 0.973
P(AC) = 1 - 0.973 = 0.027
That is, about a 3% chance that in a group of 5 people at
10 least two people share the same birthday.
0
Take group of 23 people….
Let A = event no one in group shares same birthday.
Then AC = event at least 2 people share same birthday.
P(A) = 365/365 × 364/365 × … × 343/365
= 0.493
P(AC) = 1 - 0.493 = 0.507
That is, about a 50% chance that in a group of 23 people at
10 least two people share the same birthday.
1
Take group of 50 people….
Let A = event no one in group shares same birthday.
Then AC = event at least 2 people share same birthday.
P(A) = 365/365 × 364/365 × … × 316/365
= 0.03
P(AC) = 1 - 0.03 = 0.97
That is, “virtually certain” that in a group of 50 people at
10 least two people share the same birthday.
2
Two-way Tables
And various
probabilities...
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3
Two-way table of counts
Rows: gender
M
F
All
Columns: pierced ears
N
Y
All
71
4
75
19
84
103
90
88
178
Cell Contents --
10
4
Count
Joint (“”) probabilities
Rows: gender
N
M
71
39.89
F
4
2.25
All
10
5
Columns: pierced ears
Y
All
19
90
10.67
50.56
84
47.19
88
49.44
75
103
178
42.13
57.87
100.00
Cell Contents -Count
Row conditional probabilities
Rows: gender
N
M
71
78.89
Columns: pierced ears
Y
All
19
90
21.11
100.00
F
84
95.45
4
4.55
All
10
6
88
100.00
75
103
178
42.13
57.87
100.00
Cell Contents -Count
Column conditional probabilities
Rows: gender
N
M
71
94.67
F
4
5.33
All
10
7
Columns: pierced ears
Y
All
19
90
18.45
50.56
84
81.55
88
49.44
75
103
178
100.00
100.00
100.00
Cell Contents -Count
Expected Value
Coincidences
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8
Roulette Color Bet
18 black, 18 red, and 2 green numbers
Bet on one of black or red
If correct , win $1
If wrong, lose $1
10
9
Is the bet fair?
Fair game : expected value is 0
Expected value =
sum of (outcome x prob)
Exp Val. = (+1)(18/38)+(-1)(20/38) = -2/38
Not fair since expected value is not 0.
11
0
Color Bet versus Number bet





11
1
Both have same expected value
How are the bets the same?
Long run result is same
How are they different?
Short run results can be quite different
Prob of Five Straight Losses
Color Bet = (20/38)5 = 0.04 , 4%
Number Bet = (37/38)5 = 0.88, 88%
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2
A Spectacular Coincidence ?
Many states draw four digit lottery
numbers
Several years ago Mass. and N.H. both
drew the same number on the same night
Associated Press wrote that this was a
spectacular 1 in 100 million coincidence
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3
Was Associated Press Right ?
Only if number picked is specified in
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4
advance of the draws.
Chance both pick the same pre-specified
number, for example 2963, is (1/10,000)
(1/10,000)
This is 1 in 100 million
But the match could have been on any of
10,000 possibilities
The correct analysis
First state could have picked any number
Chance the second state matches is
1/10,000
Answer for two specific states is 1/10,000
But there were 15 states doing this almost
every night .
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5
The prob that the 15 states all differ
First state can be any number
Prob second state differs = 9,999/10,000
Prob third state is unique = 9,998/10,000
And so on, for 15 states
Multiply these prob.'s to get probability
that all 15 differ
Answer is about 0.99 that all picked
different numbers
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6
Prob at least two states are same
Opposite from all different
Prob at least two the same = 1-Prob(all
differ)
1 - 0.99 = 0.01
About 1 in 100 ; a far cry from 1 in 100
million
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7