I do not play Lotto … the chances to win is too small In everyday conversation, what does the term “probability” measure? Mr president, the chances that sales will decrease if we increase prices are high What is the chance that the new investment will be profitable? How likely is it that the project will be finished on time? 3.1 Background(p88)
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I do not play Lotto … the chances to win is too small In everyday conversation, what does the term “probability” measure? Mr president, the chances that.
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I do not play Lotto … the chances to win is too small
In everyday conversation, what does the term “probability” measure?
Mr president, the chances that sales will decrease if we increase prices are high
What is the chance that the new investment will be
profitable?
How likely is it that the project will be finished on time?
3.1 Background(p88)
“In everyday conversation, the term probability is a measure of one’s belief in the occurrence of a future event”
NEW YORK, Mon: Mr. Webster Todd, Chairman of the American National Transportation Safety Board, said today that the chances of two jumbo jets colliding on the ground were about 6 million to one... –AAP
Professor Speed, who had strong research interests in probability, was intrigued by this statement and wondered how the board had calculated their figure. Speed wrote to the chairman. In the reply it was stated that the figure (6 million to one) has no statistical validity nor was it intended to be a rigorous probability statement.
3.1 Background(p88)
“…. Six million to one”
This is just a numerical measure of the very small likelihood of the event to occur … called a probability.
0
Impossible
1
CertainEqual likely
Probability is a
numerical measure of the
likelihood that an event will occur
3.1 Background(p88)
Deterministic versus Random experiment
3.1 Background (p88)
Statistics
Descriptive Statistics Inferential Statistics
3.1 Background(p88)
InferenceProbability Theory
Randomness
Experiment:
Toss a coinSelect a part for inspectionConduct a sales callRoll a diePlay a football game
The sample space is the set of ALL possible outcomes: S
Definition: sample space(p89)
e.g. S = {1, 2, 3, 4, 5, 6}
3.1 Background(p88)
E1: Observe a 1
E2: Observe a 2
E3: Observe a 3
A= Observe an odd number
E4: Observe a 4
E5: Observe a 5
E6: Observe a 6
Simple events: They cannot be decomposed - can have one and only one sample point
Each outcome is equally likely
}5,3,1{},,{ 531 EEEA
Experiment: Roll a die
3.2 First Principles(p90)
P(outcome) = 1/N
N = total number of outcomes of the experiment
P(1) = P(2) = … P(6) = 1/6
P(event) = (# outcomes in the event)
NP(A) = 3/6
3.2 First Principles(p90)
3.2 First Principles(p90)
P( ) Number of “pink” plants
Total number of plants
= 4/12 = 1/3
3.2 First Principles(p90)
3.2 First Principles(p90): Example 3.1 (p91)
Subject number
Gender Age
1 M 40 2 F 42 3 M 51 4 F 58 5 M 67 6 F 70
Event
A = Female subjectsB = Male subjectsC = subjects over the age of 65
P(A) = P(B) = P(C) =
3/6
3/6
2/6
3.2 First Principles: Some rules and concepts(p91 – p95)
Complement rule
The union of two events
The intersection of two events
The additional rule
Mutually exclusive
The conditional probability rule
Independent events
A graphic technique for visualizing set theory concepts using overlapping circles and shading to indicate intersection, union and complement.
It was introduced in the late 1800s by English logician, John Venn, although it is believed that the method originated earlier.
3.2 First Principles: Some rules and concepts(p91 – p95)
Is an insect Hatches from an egg Compound eyes Six legs Two pairs of wings Wings straight
above when at rest Thin hairless body Have a knob at the
end of the antennae
Is an insect Hatches from an egg Compound eyes Six legs Two pairs of wings Wings like a tent or
flat when at rest Wide furry body Antennae are thick
and furry
Set:“B”
Set:“M”
Elements of set B Elements of set M
3.2 First Principles: Some rules and concepts(p91 – p95)
•Is an insect•Hatches from an egg•Compound eyes•Six legs•Two pairs of wings•Wings straight above when at rest•Thin hairless body•Have a knob at the end of the antennae
•Is an insect•Hatches from an egg•Compound eyes•Six legs•Two pairs of wings•Wings like a tent or flat when at rest •Wide furry body•Antennae are thick and furry
•Wings straight above when at rest•Thin hairless body•Have a knob at the end of the antennae
•Wings like a tent or flat when at rest •Wide furry body•Antennae are thick and furry
•Is an insect•Hatches from an egg•Compound eyes•Six legs•Two pairs of wings
3.2 First Principles: Some rules and concepts(p91 – p95)
3.2 First Principles: Some rules(p91)
Subject number
Gender Age
1 M 40 2 F 42 3 M 51 4 F 58 5 M 67 6 F 70
Event
A = Female subjectsB = Male subjectsC = subjects over the age of 65
2
46
A1
35
BC
S
3.2 First Principles: Some rules(p91)
The Complement of an event:
= all outcomes in the sample space that are not in the event
S
)(1)()( APAPAP A
24
6
A 1
35
BC
S
)(CP 4/6
“At least one occurs”
A or B
The union of A and B is the event containing all sample points belonging to A or B or both.
The intersection of A and B is the event containing the sample points belonging to both A and B.
“AND”
“Both events occur”
3.2 First Principles: Some rules(p91)
3.2 First Principles: Some rules(p91)
24
6
A 1
35
BC
S
4/6P(Female or over the age of 65) = )( CAP
P(Female and over the age of 65) = )( CAP 1/6
?)( BAP0)(
P
BA
Two events are said to be mutually exclusive if they have no outcomes in common
3.2 First Principles: Some rules(p91)
The additional rule
24
6
A 1
35
BC
S
)()()( BPAPBAP
)()()()( CAPCPAPCAP
?)( CAP
The probability of rain today (mid February) is 0.6
It has been raining the whole week.
The probability of rain today (mid February) ?????
3.2 First Principles: Some rules(p91)
Conditional probability
1/6
If we know that an odd number has fallen …
P(“1”) = 1/3
Conditional Probability
P(“1”) =
3.2 First Principles: Some rules(p91)
Conditional probability
3.2 First Principles: Some rules(p91)
150
80
70
50 70 30
List all possible outcome of the one event
List all possible outcome of the other event
The sample space
3.2 First Principles: Contingency Table or cross tabulation(p93)
A two-way frequency distribution of 220 persons employed by a specific research institution, classified according to type of post and gender is given in the table below:
Calculate the probability that a randomly chosen employee:
a. Is male
b. Is a female researcher
c. is a female, given that the employee has a management post
P(M)=96/220
P F R( ) 80
220
3.2 First Principles: Contingency Table or cross tabulation(p93)
Educational level of patients seeking care at an allergy clinic
Are the two events “Male” and “17+” independent?
3.2 First Principles: Independent: Example 3.2B(p95)
210
25
210
50
210
105)17()(
210
25)17(
PMaleP
MaleP
Self study: Example 3.3
1 3 52 4 6
3.3 Combinations and permutations (p98)
Sampling With replacement Sampling Without replacement
Order important
Order not important
Order important
Order not important
1 1
1 2
2 1
1 1
1 2
1 1
1 2
2 1
1 1
1 2
3.3 Combinations and permutations (p98)
Sampling With replacement
Sampling Without replacement
Order important
Order not important
Order important
Order not important
1
23
1 23
3.3 Combinations and permutations (p98)
ABCACBBCABACCABCBA
ABC
3.3 Permutations (p99)
When sampling WITHOUT replacement, the number of distinct
arrangements (i.e., order important), called permutations of n
individuals from a population of N, is given by
)!(
!
nN
NPnN
N! = N(N-1)(N-2) … (3)(2)(1)0!0! = 1
3.3 Combinations (p100)
When sampling WITHOUT replacement, the number of samples
in which order is not important, or combinations, of n
individuals from a population of size N is given by
)!(!
!
nNn
NCnN
3.3 Permutations and Combinations (p99)
Definition of a Random variable:
A random variable is a numerical description of the
outcome of an experiment
Experiment Outcomes
Numerical Description = Random variable
3.3 Random Variable (p100)
3.3 Probability Distribution(p102)
Sampling Without replacement Order not important
1 3 52 4 6
Let X = number of females
X P(X)
0
1
2
3/15
1
9/15
3/15
1 1 1
1
0 0
1
1 1
1
1
0
2
22
The values of the random variable and the corresponding probabilities constitutes a probability distribution
The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.
Consider the experiment of tossing a coin twice and noting the outcome after every toss. Let X = the number of heads
The probability distribution of X:
X P(X)
0
1
2
1/4
1
2/4
1/4
H H
H T
T
T T
H
3.3 Probability Distribution(p102)
1 2
The probability distribution of X:
X P(X)
012
1/4
1
2/41/4
H HH TTT T
H
0 1 20
0.1
0.2
0.3
0.4
0.5
0.6
P(x)
210for )5.0()5.0(2
)( 2 ,,xx
xXP xx
The probability distribution for a discrete variable Y can be represented by a table or a graph or a formula.
3.3 Probability Distribution(p102)
A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2 or 3. Let X be a random variable indicating the number of sessions required to gain the patient’s trust. The following probability function has been proposed:
3 ,2,1for 6
)( orxx
xXP
a. Is this probability distribution valid? Explain.b. What is the probability it takes exactly two sessions
to gain the patient’s trust?c. What is the probability it takes at least two sessions