Lecture 5
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Probability
The concept of probabilities is the basis of statistical analysis
What is a probability? How is it derived?
Probability is a very familiar concept to most people but it is also a strange concept because its interpretation or meaning varies depending on the context
Example 1
What is the probability of heads for a coin toss?
Where does this figure come from?
Example 2
What is the probability that UCONN women’s BB will win the national tournament this season?
Where do we get this probability figure?
How do the two examples differ?
There are two common interpretations of probabilityObjective interpretation
Subjective interpretation
The interpretation that is used depends on which is applicable to the situation
Objective Interpretation
Applies to repeatable situations such as a coin toss
We can toss the coin many times and count the percentage of time we get heads - the probability of getting heads is simply the relative frequency of heads for a large number of tosses
Objective versus Subjective Interpretations
Subjective Interpretation
Applies to non-repeatable situations such as chance of UCONN winning
In this case, we frequently rely on the subjective interpretation - personal judgement
When the weather person says that there is a 90% chance of rain is that objective or subjective?
Basic Probability Concepts
Concepts and terminology used in studying probability
experiments, sample points, sample space, events
Experiment
An activity with more than one outcome - before the activity is performed we do not know which outcome will occur
Sample points and Sample Space
Example 1: We are conducting a survey in which two students will be selected at random - the variable of interest is whether or not the students are from Connecticut
The
Sample Points are the possible outcomes or results for this experimentGive one sample point for the example problem
CN - first student from Conn, second non-Conn
Sample SpaceThis is the set of ALL possible sample points
What is the sample space for the example problem?
SS~ CC, CN, NC, NN
Univariate or Multivariate Sample Space?
Example 2: We are conducting a survey in which two students will be selected at random - the variable of interest is the number of students from Connecticut
Sample points?
Sample Space?
SS~ 0, 1, 2
Univariate or Bivariate Sample Space
Is the sample space univariate for example 1?
No, each outcome contains results for TWO variables
Example 2?
Yes, univariate
Display of Sample Space
Tabular Display or Venn Diagram
Example one student selected at random. Variables of interest are a) gender, b) residency (Connecticut or not)
Bivariate? One basic outcome? Sample space?
Tabular Display (used for bivariate sample space)
F M Conn o1 o2 non-Conn o3 o4
Events
Events
Defined as a sub-set of the sample-space
Example: One student selected - the variables are gender and residency
We can define an event, E, which is the event that the student is from Connecticut
E occurs if o1 or o2 occurs
E* is the complement of E
E* consist of all sample points not in E
E* occurs if E does not occur
Venn Diagram
F M Conn o1 o2 non-Conn o3 o4 E1 - from Connecticut.............E2 - Female
Calculating Probabilities
F M Conn o1 o2 non-Conn o3 o4 Give: P(o1) = 0.20.............P(o2) = 0.60............. P(o3)=0.10.............P(o4)=0.10
From Venn Diagram
P(E1) = P(o1) + P(o2) = 0.80
Lecture 6
Intercept and Union
Example:
Two variables - type and condition of roadsSample points?
concrete asphalt composite Good o1 o2 o3 Fair o4 o5 o6 Bad o7 o8 o9 If E1 is the event, road is in bad condition, what are the sample points for E1?
If E2 is the event, road is asphalt, what are the sample points for E2?
Intercept of E1 and E2
Both events occurred - the .AND. operator
Union of E1 and E2
Either E1 or E2 or both occurred - the .OR. operator
Mutually Exclusive Events
Mutual Exclusive Events
Two events that cannot occur at the same time
Are E and E* mutually exclusive?
E1 - road is bad E2 - road is asphalt
E3 - road is fair E4 - road is good
E5 - road is concrete
Are E1 and E2 mutually exclusive?
Are E3 and E4 mutually exclusive?
Class Quiz
Are E1 and E5 m.e.?
What sample points constitute E1*?
Probability Distributions
Probability Distribution give the probability for each sample point in the sample space
P(oi) always lie between zero and one
Example: Univariate Probability Distribution
Road Type P(Ai) A1: concrete 0.30 A2: asphalt 0.40 A3: composite 0.30
Bivariate Probability Distributions
Example: Bivariate Probability Distribution
A1:concrete A2:asphalt A3:composite Total B1:Good 0.20 0.20 0.15 0.55 B2:Fair 0.05 0.10 0.05 0.20 B3:Bad 0.05 0.10 0.10 0.25 Total 0.30 0.40 0.30 1.00 From the bivariate probability distribution we can determine the JOINT probabilities and the MARGINAL probabilities
Joint Probability - is the probability for the joint outcome of two or more events
Joint probability of A1 and B3 = 0.05
Marginal Probability
Marginal Probability gives the probability distribution for each variable
Example: Marginal Probability for Road Type
Probability Theorems
Two basic theorems
Addition Theorem - used to find the UNION of events
Multiplication Theorem - used to find the INTERCEPT of events
Example: Student selected at random from class - we are interested in gender and place of residency
| A1:Female | A2:Male | Total | |
| B1:Conn | 2/36 | 29/36 | 31/36 |
| B2:non-Conn | 0/36 | 5/36 | 5/36 |
| Total | 2/36 | 34/36 | 36/36 |
Addition Theorems
What is the probability of the union of A1 and B1?
Addition theorem gives formula for this probability
What is the probability of A1 union B1?
Venn Diagram
Complementary Theorem
P(E1*) = 1 - P(E1)
Lecture 7
Example: Students selected at random from class - we are interested in gender and place of residency
Conditional Probability
Results for CE251 in 1993
Total number of students - 36
Number of females (Event A1) - 2
Total number from Connecticut (Event B1) - 31
Number of males from Connecticut - 29
Develop the joint probability distribution for this example
What is the probability the student selected is female?
Conditional Probability
If the student is female, what is probability that the student is from out of state?
This last is a CONDITIONAL probability
The probability that the student is from out of state GIVEN that the student is female is written as
P(B2|A1) - conditional probability
Determine P(B2|A1) and P(A1|B1)
Are A1 and B2 m.e.?
Derive the formula for P(B2|A1)
Multiplication Theorems
What is the probability of the intercept of A1 and B1?
Multiplication theorem gives formula for this probability
can be derived from the conditional probability formula
Relationships between Variables
In many statistical studies we want to know if there is a relationship between the variables in the study
Are the variables Dependent or Independent?
If they are dependent, to what degree are they dependent?
Example: Evaluation of a Medical Test
Two variables
Test results - Positive(A1) or Negative (A2)
Disease Status - Have disease(B1) or Not have disease(B2)
Should the variables be independent or dependent?
If the variable are independent what does that tell us about the efficacy of the test
Independent Variables
Independent - there is no relationship between the two variables. The probability of one is NOT CONDITIONAL on the other
Probability of positive test result is independent of whether or not the person has the disease
In other words, for independent variable
P(A1) = P(A1|B1) = P(A1|B2)
This is the statistical test of independence
Example: Is there a relationship between Soil type and Soil Strength
| Soil Strength | B1:A-1-b | B2:A-2-4 | B3:A-4 | Total |
| A1:High | 0.08 | 0.02 | 0.17 | 0.27 |
| A2:Medium | 0.04 | 0.20 | 0.21 | 0.45 |
| A3:Low | 0.04 | 0.08 | 0.25 | 0.37 |
| Total | 0.17 | 0.20 | 0.63 | 1.00 |
AASHTO Soil Classifications used
Does the results indicate a relationship between type and strength?
Hard to see from prob. dist. - need conditional probs
Independent or Dependent
= 0.08/0.17 = 0.50
Condition on Soil Strength B1:A-1-b B2:A-2-4 B3:A-4 A1:High 0.50 0.10 0.27 A2:Medium 0.25 0.50 0.33 A3:Low 0.25 0.40 0.40 Is there a relationship between soil type and strength?
What is the relationship?
Joint Probability for Independent Events
for independence P(
B1|A1) = P(B1)therefore
This is a very useful result - if two events are independent then we can easily find the joint probability of the two events
Class Exercise
Experiment consist and flipping coin twice and recording heads or tails for each flip
Develop Theoretical Prob. Distribution
Do experiment once, twice, trice for each person
Bayes’ Theorem
Bayes’ Theorem is an extension of the conditional probability formula and is used for many engineering and management problems
Bayes’ theorem is used in situations where we have an initial probability assessment for an event - if we get additional information we can re-assess our probability using Bayes’ formula
Initial Probability Assessment - Prior Probability
Revised Probability Assessment - Posterior Probability
UCONN EIT Results
Bayes’ Theorem Example
We want to assess the chances of a UCONN student passing the EIT
We know that 80% of students taking the EIT pass the test
We can use this as the PRIOR probability for UCONN students
We find out that
40% of students who pass are UCONN students
10% of students who fail are UCONN students
Based on these facts we can revise the probability that UCONN students will pass
This revised probability is the POSTERIOR probability
Passing the EIT!
Calculating the chance of a UCONN student passing the EIT
Define the events
A1 - student pass A2 - student fail
B1 - student from UCONN
B2 - student is non-UCONN
Rewrite the information in probability notations
Prior probability -
probability assessment before new infoAdditional Information
40% of passing students from UCONN
10% of failing students from UCONN
Posterior Probability -
What do we want to know?
Passing the EIT!
Develop Probability Distribution for EIT Problem?
We need P(A1|B1)!
Formula for P(A1|B1)?
what is (from multiplication theorem)?
what is P(B1)?
If we substitute these into the condition probability formula we get the Bayesian Equation
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Date of last update - 16 Sep 1998