This presentation will clarify all your basic concepts of Probability. It includes Random Experiment, Sample Space, Event, Complementary event, Union - Intersection and difference of events, favorable cases, probability definitions, conditional probability, 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
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
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
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
3 PROBABILITY TOPICSFigure 3.1 Meteor showers are rare, .docxtamicawaysmith
3 | PROBABILITY TOPICS
Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr)
Introduction
Chapter Objectives
By the end of this chapter, the student should be able to:
• Understand and use the terminology of probability.
• Determine whether two events are mutually exclusive and whether two events are independent.
• Calculate probabilities using the Addition Rules and Multiplication Rules.
• Construct and interpret Contingency Tables.
• Construct and interpret Venn Diagrams.
• Construct and interpret Tree Diagrams.
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess
their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can
comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You
may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You
may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals
with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study
for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic
approach.
Your instructor will survey your class. Count the number of students in the class today.
• Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.
CHAPTER 3 | PROBABILITY TOPICS 163
• Raise your hand if you rode a bus within the past month. Record the number of raised hands.
• Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.
Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen
person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person
in your class rode a bus within the last month and so on. Discuss your answers.
• Find P(change).
• Find P(bus).
• Find P(change AND bus). Find the probability that a randomly chosen student in your class has change in his/her
pocket or purse and rode a bus within the last month.
• Find P(change|bus). Find the probability that a randomly chosen student has change given that he or she rode a
bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus,
count those who have change. The probability is equal to those who have change and rode a bus divided by those
who rode a bus.
3.1 | Terminology
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.
An e ...
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
3 PROBABILITY TOPICSFigure 3.1 Meteor showers are rare, .docxtamicawaysmith
3 | PROBABILITY TOPICS
Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr)
Introduction
Chapter Objectives
By the end of this chapter, the student should be able to:
• Understand and use the terminology of probability.
• Determine whether two events are mutually exclusive and whether two events are independent.
• Calculate probabilities using the Addition Rules and Multiplication Rules.
• Construct and interpret Contingency Tables.
• Construct and interpret Venn Diagrams.
• Construct and interpret Tree Diagrams.
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess
their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can
comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You
may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You
may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals
with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study
for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic
approach.
Your instructor will survey your class. Count the number of students in the class today.
• Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.
CHAPTER 3 | PROBABILITY TOPICS 163
• Raise your hand if you rode a bus within the past month. Record the number of raised hands.
• Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.
Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen
person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person
in your class rode a bus within the last month and so on. Discuss your answers.
• Find P(change).
• Find P(bus).
• Find P(change AND bus). Find the probability that a randomly chosen student in your class has change in his/her
pocket or purse and rode a bus within the last month.
• Find P(change|bus). Find the probability that a randomly chosen student has change given that he or she rode a
bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus,
count those who have change. The probability is equal to those who have change and rode a bus divided by those
who rode a bus.
3.1 | Terminology
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.
An e ...
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2. History
• History of Probability goes back in mid sixteenth
century around 1654.
• Chevalier de Méré, a French nobleman with an
interest in gaming and gambling questions, called
Pascal’s and Fermat’s attention to an apparent
contradiction concerning a popular dice game.
3. The Problem of Points
• Question posed by a gambler, Chevalier De Mere and then discussed by Pascal
and Fermat.
– There are many versions of this problem, appearing in print as early as 1494
and discussed earlier by Cardano and Tartaglia, among others.
• Two players of equal skill play a game with an ultimate monetary prize. The first
to win a fixed number of rounds wins everything.
• How should the stakes be divided if the game is interrupted after several rounds,
but before either player has won the required number?
4. Basic terms
• Random Experiment
• Sample Space
• Event
• Complimentary event
• Union of events
• Interaction of events
• Difference of events
• Mutually exclusive events
• Exhaustive events
• Equally likely events
• Independent
• Favourable events
5. Random Experiment
An experiment which can result in any one of several possible
outcomes is called a random experiment.
Characteristics of a random experiment
• The experiment results in any one of the several outcomes
• All possible outcomes of the experiment can be described in
advance but cannot be known in advance
• The experiment can be repeated under same conditions
6. Sample space
A set of all possible outcomes of a random experiment is called a
sample space.
E.g. Consider an experiment of throwing a dice,
then the sample space, S = { 1,2,3,4,5,6}.
E.g. Consider an experiment of result of a student in an exam,
then the sample space, S = {Pass, Fail}
E.g. Consider an experiment of throwing two coins,
then the sample space, S = {HH, HT, TH,TT}
7. Event
The result of a random experiment is called an event. or
A subset of a sample space is called an event.
E.g. If S = {1,2,3,4,5,6} then event A of getting an even number is
A = {2, 4, 6}
E.g. If S = {HH,HT,TH,TT} then event B of getting atleast one tail is
B = {HT,TH,TT}
E.g. If S= {1,2,3…100} the event C of getting a number divisible by 3 is
C = {3,6,9,12,…99}
8. Complementary event
The set of all the sample points of sample space which do not
belong to A, is called the complement of an event A.
E.g. If S = {1,2,3,4,5,6} and A = {2,4,6} then A𝑐 = {1,3,5}.
9. Union of events
The set of all sample points belonging to either A or B or both is
called the union of events A and B. It is denoted by or
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A U B = {1,2,3,4,5,6,8,10}
Theevent that either A or B or both A and B occurs is known as
the union of two events A and B.
A B
.
10. Intersection of events
The set of all sample points belonging to A and B is called the
intersection of events A and B. It is denoted by or
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A ∩ B = {2,4,6}
The event that both A and B occurs together is known as the
intersection of two events A and B.
A B
.
11. Difference of events
The set of sample points belonging to A but not in B is called the
difference of events A and B. It is denoted by A - B or
If the event A occurs, event B does not occur, then it is called the
difference of events A and B.
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A − B = {1,3,5} and B – A = {8,10}
12. Exhaustive
If all possible outcomes of an experiment are considered, the
outcomes are called to be exhaustive.
E.g. If S= {1,2,3,4,5,6} , A= {1,2,3,5} and B = {2,4,6}
then A and B are called exhaustive as A U B = S
13. Mutually exclusive
If the occurrence of one event prevents the occurrence of
remaining events then events are called mutually exclusive
events. or
If events cannot occur together then they are called mutually
exclusive events.
In other words A ∩ B = Ø
E.g. A person cannot be of age 15 years and
also be voter of India.
14. Equally likely
If the events have equal chance of occurrence, events are called
equally likely events. or
When none of the outcome is expected to occur in preference to
another, such events are known as equally likely events.
E.g. In tossing of coin, outcomes Head or Tail are expected to occurs same
number of times.
E.g. In throwing a dice, chance of getting number 1 or 2 or 3 or 4 or 5 or 6 are
expected to be same.
15. Independent events
If the occurrence of one event does not depend on occurrence or
non-occurrence of other events then events are called
independent events.
E.g. In an experiment of tossing a coin and throwing a dice. Event of
getting Head on a coin does not depend on getting even number or odd
number on dice.
16. Independent events
E.g. Three persons say Rahul, Narendra and Akhilesh aim a target.
We know the chance of Narendra hitting the target does not depend on
whether Rahul or Akhilesh hits the target or misses the target.
E.g. Two students of HL, one of Section 2 and other of Section 4 are given
a statistics problem to solve. We know the chance of Section 2 student
solving the problem does not depend whether student of Section 4 solves
or does not solves the problem.
E.g. For a couple if husband dies can we surely say wife will also die. She
may or she may not.
17. Favourable cases
The sample points in the sample space that helps in the occurrence of
an event are called favourable cases.
E.g. If S = {1,2,3,4,5,6} and event A is even number
then favourable cases are {2,4,6}.
E.g. If a card is selected from a pack of 52 cards and the event B is getting an Ace
card then favourable cases { Club Ace, Heart Ace, Diamond Ace, Spade Ace}
18. Mathematical or Classical or
Apriori definition of Probability
If the experiment can result in 𝑛 exhaustive, mutually exclusive
and equally likely ways and if 𝑚 of them are favourable to the
occurrence of an event A, then the probability of occurrence of
an event A is denoted by P(A).
i.e. P(A) =
𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑐𝑎𝑠𝑒𝑠
=
𝑚
𝑛
19. Mathematical or Classical or
Apriori definition of Probability
E.g. Consider an experiment of throwing a dice. Find probability of getting
number greater than 4.
Here, sample space: S= {1,2,3,4,5,6} then favourable cases for the event A
of getting number greater than 4 are { 5,6}, so
P(A) =
𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑐𝑎𝑠𝑒𝑠
=
𝑚
𝑛
=
2
6
= 0.33
20. Statistical or Empirical or
Aposteriori definition of Probability
If an experiment is repeated under essentially, the same
conditions for a large number of times then the limit of the ratio
of number of times the event occurs to the total number of trails
is defined as the probability of the event.
i.e. P(A) = lim
𝑛→∞
(
𝑚
𝑛
)
21. Modern or Axiomatic definition of Probability
If P(A) is the real number assigned to the subset A of the sample
space S, then it is called the probability of an event A, provided
P(A) satisfies the following postulates:
Postulate 1: 0 ≤ 𝑃 𝐴 ≤ 1
Postulate 2: P(S)=1
Postulate 3: If 𝐴1, 𝐴2, 𝐴3,… is finite or infinite sequence of
mutually exclusive events i.e. subsets of S then
𝑃(𝐴1 ∪ 𝐴2 ∪ 𝐴3 ∪……) = P(𝐴1)+P(𝐴2)+P(𝐴3)+…..
22. Conditional Probability
The probability of occurrence of an event B, when A has occurred
is called conditional probability of B under A and it is denoted by
P(B/A).
P(B/A) =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
; P(A)≠ 0
23. Bayes Theorem
If an event D can occur with mutually exclusive and exhaustive events
𝐴1, 𝐴2, 𝐴3,…..𝐴 𝑛 and if the probabilities P(𝐴1), P(𝐴2), P(𝐴3),…..P(𝐴 𝑛)
and also the conditional probabilities P(D/𝐴1), P(D/𝐴2), P(D/𝐴3), …….
P(D/𝐴 𝑛) are known then the inverse probability of occurrence of an
event 𝐴𝑖 under the condition that the event D has occurred is given
by
P(𝐴𝑖/ 𝐷) =
𝑃(𝐴 𝑖∩𝐷)
𝑃(𝐷)
=
P( 𝐴 𝑖)P(D/ 𝐴 𝑖)
P( 𝐴1)P(D/ 𝐴1)+P( 𝐴2)P(D/ 𝐴2)+P( 𝐴3)P(D/ 𝐴3)+..…P( 𝐴 𝑛)P(D/ 𝐴 𝑛)