This document provides an overview of key concepts in probability, including classical, frequentist, and subjective definitions of probability. It discusses sample spaces, events, mutually exclusive and independent events, and the rules of addition, multiplication, total probability, and Bayes' theorem. It also covers applications of probability, such as screening tests, and how concepts like sensitivity, specificity, and predictive values are used to evaluate screening tests using Bayesian reasoning.
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
Mathematics, Statistics, Probability, Randomness, General Probability Rules, General Addition Rules, Conditional Probability, General Multiplication Rules, Bayes’s Rule, Independence
Mathematics, Statistics, Probability, Randomness, General Probability Rules, General Addition Rules, Conditional Probability, General Multiplication Rules, Bayes’s Rule, Independence
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
1 Probability Please read sections 3.1 – 3.3 in your .docxaryan532920
1
Probability
Please read sections 3.1 – 3.3 in your textbook
Def: An experiment is a process by which observations are generated.
Def: A variable is a quantity that is observed in the experiment.
Def: The sample space (S) for an experiment is the set of all possible outcomes.
Def: An event E is a subset of a sample space. It provides the collection of outcomes
that correspond to some classification.
Example:
Note: A sample space does not have to be finite.
Example: Pick any positive integer. The sample space is countably infinite.
A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that
has a countably infinite number of elements { }1,3,5,7,... .
A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <
2
A Venn diagram is used to show relationships between events.
A intersection B = (A ∩ B) = A and B
The outcomes in (A intersection B) belong to set A as well as to set B.
A union B = (A U B) = A alone or B alone or both
Union Formula
For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e.
P (A U B) = P (A) + P (B) – P (A ∩ B)
3
cA complement not A A ' A A = = = =
A complement consists of all outcomes outside of A.
Note: P (not A) = 1 – P (A)
Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect,
i.e. if they do not occur at the same time. They have no outcomes in common.
When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.
Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B)
(This is exactly the same statement as rule 3 below)
Axioms of Probability
Def: A probability function p is a rule for calculating the probability of an event. The
function p satisfies 3 conditions:
1) 0 ≤ P (A) ≤1, for all events A in the sample space S
2) P (Sample Space S) = 1
3) If A, B, C are mutually exclusive events in the sample space S, then
P(A B C) P(A) P(B) P(C)∪ ∪ = + +
4
The Classical Probability Concept: If there are n equally likely possibilities, of which one
must occur and s are regarded as successes, then the probability of success is s
n
.
Example:
Frequency interpretation of Probability: The probability of an event E is the proportion of
times the event occurs during a long run of repeated experiments.
Example:
Def: A set function assigns a non-negative value to a set.
Ex: N (A) is a set function whose value is the number of elements in A.
Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and
B are mutually exclusive.
N (A) is an additive set function.
Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the
event that the sum on ...
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
➽=ALL False flag-War Machine-War profiteering-Energy (oil/Gas) Iraq, Iran,…oil and gas
USA invades other countries just to own their natural resources and to place them in the hands of American corporations. Facebook doesn’t call that terrorism. They call it democracy. BBC, CNN, FOX NEWS, FR 24, ITV/CH 4, SKY, EURO NEWS, ITV trash Sun paper,… Facebook all are protector and preserver of the propaganda classifying IR Iran as a dangerous terrorist organization. But FB, BBC, CNN, FOX NEWS, FR 24, ITV/CH 4-SKY, EURO NEWS, ITV do know well, that USA is the biggest terrorist country in the world.
‘terrorism’ the unlawful use of violence and intimidation, especially against civilians, in the pursuit of political aims.
"the fight against terrorism" is the fight against the unlawful use of violence and intimidation and carpet bombing.
Ever since the beginning of the 19th century, the West has been sucking on the jugular vein of the Moslem body politic like a veritable vampire whose thirst for Moslem blood is never sated and who refused to let go. Since 1979, Iran, which has always played the role of the intellectual leader of the Islamic world, has risen up to put a stop to this outrage against God’s law and will, and against all decency.
MY NEWS PUNCH DR F DEJAHANG 28/12/2019
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NEWS YOU WON’T FIND ON BBC-CNN-FOX NEWS, FR 24, EURO NEWS, ITV…
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Articles for Political Science, Mathematics and Productivity the Student Room BSc, MSc & PhD Project Mangers etc
PPTs in SLIDESHARE International Studies Research Degrees (MPhilPhD)
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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1. 1
516 Probabilty Review
0
Probability
• Probability - meaning
1) classical
2) frequentist
3) subjective (personal)
• Sample space, events
• Mutually exclusive, independence
• and, or, complement
• Joint, marginal, conditional probability
• Probability - rules
1) Addition
2) Multiplication
3) Total probability
4) Bayes
• Screening
•sensitivity
•specificity
•predictive values
1
Probability
Probability provides a measure of uncertainty associated with the
occurrence of events or outcomes – Models and theory
Definitions: 1. Classical: P(E) = m/N
If an event can occur in N mutually exclusive, equally
likely ways, and if m of these possess characteristic E,
then the probability is equal to m/N.
Example: What is the probability of rolling a total of 7 on two dice?
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516 Probabilty Review
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2. relative frequency:
If a process or an experiment is repeated a large number of
times, n, and if the characteristic, E, occurs m times, then the
relative frequency, m/n, of E will be approximately equal to the
probability of E.
P(E) m / n≈
3. personal probability
What is the probability of life on Mars?
» Around 1900, the English statistician Karl Pearson heroically tossed a coin
24,000 times and recorded 12,012 heads, giving a proportion of 0.5005.
probabilityofhead
n
0 500 1000
0
.5
1
Stata: graph twoway line prob n,
yscale(r(0.0,1.0)) title("Probability
of heads") ylabel(0(0.5)1)
3
Sample Space
The sample space consists of the possible outcomes of an
experiment. An event is an outcome or set of outcomes.
For a coin flip the sample space is (H,T).
3. 3
516 Probabilty Review
4
1. Two events, A and B, are said to be mutually exclusive (disjoint) if
only one or the other, but not both, can occur in a particular
experiment.
2. Given n mutually exclusive events, E1, E2, …., En, the probability of any
event is non-negative and less than or equal to 1:
3. The sum of the probabilities of an exhaustive collection (i.e. at least
one must occur) of mutually exclusive outcomes is 1:
4. The probability of all events other than an event A is denoted by P(Ac)
[Ac stands for “A complement”] or [“A bar”]. Note that
P(Ac) = 1 – P(A)
Basic Properties of Probability
0 P ( ) 1iE≤ ≤
P(E ) P(E ) P(E ) P(E ) 1i
i 1
n
1 2 n
=
∑ = + + + =K
BA
A Ac
P( )A
5
Notation for Joint Probabilities
• If A and B are any two events then we write
P(A or B) or P(A ∪ B)
to indicate the probability that event A or event B (or both) occurred.
• If A and B are any two events then we write
P(A and B) or P(AB) or P(A ∩ B)
to indicate the probability that both A and B occurred.
• If A and B are any two events then we write
P(A given B) or P(A|B)
to indicate the probability of A among the subset of cases in which B
is known to have occurred.
BA
A ∩ B is the overlap
A ∪ B is the
entire shaded
area
4. 4
516 Probabilty Review
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Conditional Probability
The conditional probability of an event A given B (i.e.
given that B has occurred) is denoted P(A | B).
Disease Status
Pos. Neg. Total
Pos. 9 80 89Test
Result Neg. 1 9910 9,911
Total 10 9990 10,000
What is P(test positive)?
What is P(test positive | disease positive)?
What is P(disease positive | test positive)?
7
5. 5
516 Probabilty Review
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General Probability Rules
• Addition rule
If two events A and B are not mutually exclusive, then the probability
that event A or event B occurs is:
E.g. Of the students at Anytown High school, 30% have had the
mumps, 70% have had measles and 21% have had both. What is the
probability that a randomly chosen student has had at least one of the
above diseases?
P(at least one) = P(mumps or measles)
= .30 + .70 - .21 = .79
P(AB)P(B)P(A)B)orP(A −+=
Mumps
(.30)
Measles
(.70)
Both (.21)
What is P(neither)? Identify the area for “neither”
9
General Probability Rules
• Multiplication rule (special case – independence)
If two events, A and B, are “independent” (probability of one does not
depend on whether the other occurred) then
P(AB) = P(A)P(B)
Mumps?
Measles?
Measles?
P(Mumps)=0.3
P(no Mumps)=0.7
P(Measles)=0.7
P(no Measles)=0.3
Mumps,Measles
[P(both)=0.21]
Mumps, No Measles
[P(mumps only)=0.09]
No Mumps, Measles
[P(measles only)=0.49]
No Mumps, No Measles
[P(neither)=0.21]
Easy to extend for independent events A,B,C,… :
P(ABC…) = P(A)P(B)P(C)…
P(Measles)=0.7
P(no Measles)=0.3
6. 6
516 Probabilty Review
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• Multiplication rule (general)
More generally, however, A and B may not be independent. The
probability that one event occurs may depend on the other event. This
brings us back to conditional probability. The general formula for the
probability that both A and B will occur is
Two events A and B are said to be independent if and only if
P(A|B) = P(A) or
P(B|A) = P(B) or
P(AB) = P(A)P(B).
(Note: If any one holds then all three hold)
E.g. Suppose P(mumps) = .7, P(measles) = .3 and P(both) = .25. Are
the two events independent?
No, because P(mumps and measles) = .25 while
P(mumps)P(measles) = .21
P( ) P( | )P( ) P( | )P( )AB A B B B A A= =
General Probability Rules
11
• Total probability rule
If A1,…An are mutually exclusive, exhaustive events, then
1 1
P( ) P( and ) P( | )P( )
n n
i i i
i i
B B A B A A
= =
= =∑ ∑
E.g. The following table gives the estimated proportion of individuals with
Alzhiemer’s disease (AD) by age group. It also gives the proportion of
the general population that are expected to fall in the age group in 2030.
What proportion of the population in 2030 will have Alzhiemer’s disease?
Proportion
with AD
Proportion
population
< 65 .00 .80
65 – 75 .01 .11
75 – 85 .07 .07
Age
group
> 85 .25 .02
P(AD in 2030) = 0*.8 + .01*.11 + .07*.07 + .25*.02 = .011
7. 7
516 Probabilty Review
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• Bayes rule (combine multiplication rule with total probability rule)
1
P( | )P( )
P( | )
P( )
P( | )P( )
P( | )P( )
n
i i
i
B A A
A B
B
B A A
B A A
=
=
=
∑
We will only apply this to the situation where A and B have two
levels each, say, A and A, B and B. The formula becomes
P( | )P( )
P( | )
P( | )P( ) P( | )P( )
B A A
A B
B A A B A A
=
+
13
Screening - an Application of Bayes Rule
D = disease pos.
T = test pos.
Suppose we have a random sample of a population...
Disease Status
Pos. Neg. Total
Pos. 90 30 120Test
Result Neg. 10 970 980
Total 100 1000 1100
Prevalence = P(D) = 100/1100 = .091
Sensitivity = P(T | D) = 90/100 = .9
Specificity = P(T | D) = 970/1000 = .97
PPV = Predictive Value of a Positive Test
= P(D | T) = 90/120 = .75
NPV = Predictive Value of a Negative Test
= P(D | T) = 970/980 = .99
8. 8
516 Probabilty Review
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Screening - an Application of Bayes Rule
D = disease pos.
T = test pos.
Now suppose we have taken a sample of 100 disease positive and
100 disease negative individuals (e.g. case-control design)
Prevalence = ???? (not .5!)
Sensitivity = P(T | D) = 90/100 = .9
Specificity = P(T | D) = 97/100 = .97
PPV = P(D | T) = 90/93 NO!
NPV = P(D | T) = 97/107 NO!
Disease Status
Pos. Neg. Total
Pos. 90 3 93Test
Result Neg. 10 97 107
Total 100 100 200
15
Screening - an Application of Bayes Rule
D = disease pos.
T = test pos.
100
1100
100 1000
1100 1100
P( | )P( )
PPV P( | )
P( )
P( | )P( )
P( | )P( ) P( | )P( )
.9
.75
.9 .03
T D D
D T
T
T D D
T D D T D D
= =
=
+
×
= =
× + ×
Assume we know, from external sources, that P(D) = 100/1100.
NPV P( | )D T= =
9. 9
516 Probabilty Review
16
Screening for Alzheimer’s Disease
- an Application of Bayes Rule
The presence of a particular genotype, APOE, has been considered
as a screen for people who develop AD. The sensitivity and
specificity of the test are 60% and 70%, respectively.
So, there is clearly an association between APOE and AD. Those
who develop AD are twice as likely to have the genotype as those
who don’t develop AD. (To see this, compare P(APOE+|AD+) to
P(APOE+|AD-).)
But is this a useful screen for AD?
Compare 2 elderly populations:
- the general elderly population where the chance of
developing AD is 5%
- an elderly population with memory complaints where the
chance of developing AD is 70%
17
AD+
.95
.6
Screening for Alzheimer’s Disease
- an Application of Bayes Rule
Suppose we are looking at an elderly population where the chance of
developing memory complaints is 5%.
AD-
APOE+ .03
APOE-
APOE+
APOE-
.05
.4
.3
.7
.02
.285
.665
Marginal Conditional Joint
.03
PPV .095
.03 .285
= =
+
.665
NPV .97
.665 .02
= =
+
10. 10
516 Probabilty Review
18
AD+
.3
.6
Screening for Alzheimer’s Disease
- an Application of Bayes Rule
Suppose we are looking at an elderly population with memory problems
where the chance of developing memory complaints is 70%.
AD-
APOE+ .42
APOE-
APOE+
APOE-
.7
.4
.3
.7
.28
.09
.21
Marginal Conditional Joint
.42
PPV .82
.42 .08
= =
+
.21
NPV .43
.21 .28
= =
+
19
Screening for Alzheimer’s Disease
- an Application of Bayes Rule
Comments
We found that a test with given sensitivity and specificity will have
different predictive values in different populations. So the usefulness
of the test will depend on the population context and the implications of
incorrect or missed diagnoses
• If disease prevalence is low, high specificity is particularly important.
Otherwise almost all positive tests will be false positives and these
diagnoses will burden the health care system and patients with
unnecessary follow-up.
• If consequences of false positives are very bad (e.g. unnecessary
high-risk surgery), high specificity is also important.
• If disease prevalence is high and/or it is very important to find true
positive cases, then you would want a test with high sensitivity.