What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Probability Theory
01. Introduction
Shilpak Banerjee
Indian Institute of Technology Tirupati
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What is probability theory?
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Axiomatic approach to probability
References
You can use the following references for the course:
1
Introduction to probability, statistics, and random processes by
Hossein Pishro-Nik.
https://www.probabilitycourse.com/
2
Probability and Statistics I by Matthew Aldridge.
https://mpaldridge.github.io/math1710/
3
Advanced Engineering Mathematics (10th edition) by Erwin
Kreyszig.
4
Introduction to probability and statistics by Seymour Lipschutz and
John Schiller.
5
A first course in Probability (9th edition) by Sheldon Ross.
In my lectures I will be referring to the aforementioned editions.
However, you’re free to use any edition of these references that is
accessible to you.
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Experiments, Outcomes, Events
Axiomatic approach to probability
Section 1
What is probability theory?
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Coin
An Indian one rupee coin.
“Flipping” or “Tossing” a coin.
Reverse or Tails of Indian 1 Rupee Coin/Indian 1 Rupee
Coin Obverse or Heads, Author: Surya Narayan Jena,
License: CC-BY-SA-4.0, Created: 12 May 2021
Coin toss, Author: ICMA Photos, License:
CC-BY-SA-2.0, Created: 17 June 2009
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Six sided dice
Four traditional six sided dice
Four six-sided dice, Author: Diacritica, License: CC
BY-SA 3.0, Created: 6 December 2010
Result of “rolling” two dice
simultaneously.
Dice One Snake Eyes Double Roll, Author: Winter Orion,
License: CC-Zero, Created: 29 November 2015
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Other types of dice
A typical set of polyhedral dice.
A three sided die.
A 3-sided wooden die, Author: Traitor / Jacqueline de
Swart, License: CC-BY-SA-3.0-migrated, Created: 30
Six dice of various colours. 4-sided die, 6-sided die,
July 2006
8-sided die, 10-sided die, 12-sided die and 20-sided die,
Author: Diacritica, License: CC BY-SA 3.0, Created: 30
November 2010
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. What is the difference between the words “dice” and “die”?
2. Can a coin be considered as a two sided die? What about one
sided die?
3. List the games that requires you to “toss a coin” or “roll one
or more die/dice”.
4. Check out https://rolladie.net/
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
A standard deck of cards
A box containing a standard deck
of cards.
Some cards.
Bicycle Blue Standard Deck 2013 Package, Author:
Collage di Carte da gioco, Author: WhiteDr4ag0n,
MartyRus, License: CC BY-SA 4.0, Created: 30 October
License: CC-BY-SA-4.0, Created: 10 December 2014
2016
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
In many decks, the queen of clubs holds a flower. She is thus known as the "flower queen", though this design element is among the mo
variable; the Bicycle Poker deck depicts all queens with a flower styled according to their suit.
Know your deck of cards
omputer representations
VG
nicode
Example set of 52 playing cards; 13 of each suit: clubs, diamonds, hearts, and spades
Ace
2
3
4
5
6
7
8
9
10
Jack
Queen
King
Clubs
Diamonds
Hearts
Spades
Reference: https://en.wikipedia.org/wiki/Standard 52-card deck
of Unicode 7.0, playing cards are now represented. Note that the following chart ("Cards", Range: 1F0A0–1F0FF) includes cards from the Tarot Nou
k, as well as the standard 52-card deck.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. Familiarize yourself with the standard deck of cards:
1 What is a suit?
2 What is a face card or court card?
3 What is a numeral card or pip card?
2. Often a pack of standard deck of card contains additional
cards. What type of cards are those?
3. Other than the standard deck of card, what other deck of
cards can you find?
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
What does a card game look like?
Dealing cards: The cards are
distributed among the players
A hand: The cards held by one
player “playing hand”)
Author: Benebiankie, License: CC BY-SA 4.0, Created: 9
February 2019
Author: Ch Maheswara Raju, License: CC-BY-SA-4.0,
Created: 10 August 2021
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. List some games using the standard deck of cards.
2. Familiarize yourself with the terminologies associated with
some popular games using the standard deck of cards:
1
2
3
What do you mean by shuffle or shuffling the cards?
How do you deal the cards?
What is a hand?
3. Check out:
https://en.wikipedia.org/wiki/Glossary of card game terms
(No need to memorize these terms.)
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Experiments, Outcomes, Events
Axiomatic approach to probability
What does Encyclopaedia Britannica says?
“Probability theory, a branch of mathematics concerned with the
analysis of random phenomena. The outcome of a random event
cannot be determined before it occurs, but it may be any one of
several possible outcomes. The actual outcome is considered to be
determined by chance.”
Exercise
Read the article on probability theory in Encyclopaedia Britannica.
Link: https://www.britannica.com/science/probability-theory
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Axiomatic approach to probability
What does Wikipedia says?
“Probability is the branch of mathematics concerning events and
numerical descriptions of how likely they are to occur. The
probability of an event is a number between 0 and 1; the larger the
probability, the more likely an event is to occur. The higher the
probability of an event, the more likely it is that the event will
occur. A simple example is the tossing of a fair (unbiased) coin.
Since the coin is fair, the two outcomes (‘heads’ and ‘tails’) are
both equally probable; the probability of ‘heads’ equals the
probability of ‘tails’; and since no other outcomes are possible, the
probability of either ‘heads’ or ‘tails’ is 1/2 (which could also be
written as 0.5 or 50%).”
Exercise
Read the article on probability in Wikipedia.
Link: https://en.wikipedia.org/wiki/Probability
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What is probability theory?
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Axiomatic approach to probability
What does Chatgpt says?
In one sentence:
Probability theory is the mathematical study of random phenomena
using measure theory to assign and analyze probabilities of events.
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Axiomatic approach to probability
Our takeaway
There are some phenomena whose ‘outcome’ can not be predicted
in a definitive way using the given information. We still want a way
to understand how likely an outcome or certain group of outcome
will occur.
To mathematically ‘quantify’ such understanding, we can start by
listing all possible outcomes as a set, and then figure out a way to
assign a number to each outcome (or groups of outcomes)
indicating how likely that outcome (or group of outcomes) is to
occur. Higher the number, more likely it is.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Section 2
Experiments, Outcomes, Events
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Experiment, trial
An experiment is a process of
measurement or observation, in a
laboratory, in a factory, on the
street, in nature, or wherever; so
“experiment” is used in a rather
general sense.
Examples:
Inspecting a light-bulb.
Tossing a coin.
Simultaneously tossing two
coins.
Rolling a dice.
Our interest is in experiments
that involve randomness, chance
effects, so that we cannot predict
a result exactly.
A trial is a single performance of
an experiment.
Simultaneously rolling two
dice.
Measuring ambient
temperature.
Measuring the lifetime of a
transistor.
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Experiments, Outcomes, Events
Axiomatic approach to probability
Outcome, sample point, sample space
A trial is a single performance of
an experiment.
Its result is called an outcome or
a sample point.
n trials then give a sample of size
n consisting of n sample points.
The sample space S of an
experiment is the set of all
possible outcomes.
Examples:
Tossing a coin: Head (H) or
tail (T ): S = {H, T }
Simultaneously tossing two
coins:
S = {HH, HT , TH, TT }
Two trials of tossing a coin
will create a sequence to two
sample points, one for each
trial, such as any one of
HH, HT , TH, TT .
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
(E. Kreyszig: Advanced Engineering Mathematics, Examples 1-6, pp1016)
Inspecting a light-bulb:
S = {Defective, Okay}
Note: Given a light-bulb, we assume there is no randomness in
determining if it is defective or not defective. We can just plug it
in our socket and determine if it lights up or not! Where does the
randomness comes from?
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Measuring ambient temperature:
We make the assumption that the lowest temperature can be 10
degree Celcius and maximum is 45 degree Celcius. This of course
can be changed depending on where the thermometer is placed!
S = {x : 10 ≤ x ≤ 45} = [10, 45]
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Measuring the lifetime of a transistor:
Since lifetime can be any positive number in any fixed unit of time
(say seconds)
S = {t : 0 ≤ t < ∞}
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Rolling a standard six sided die:
S = {1, 2, 3, 4, 5, 6}
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Simultaneously rolling two dice:
S = {(i, j) : i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3, 4, 5, 6}
It is sometimes convenient to list the outcomes in a table (or
matrix) in some cases such as this one. We can write S =
(1, 1)
(2, 1)
(3, 1)
(4, 1)
(5, 1)
(6, 1)
(1, 2)
(2, 2)
(3, 2)
(4, 2)
(5, 2)
(6, 2)
(1, 3)
(2, 3)
(3, 3)
(4, 3)
(5, 3)
(6, 3)
(1, 4)
(2, 4)
(3, 4)
(4, 4)
(5, 4)
(6, 4)
(1, 5)
(2, 5)
(3, 5)
(4, 5)
(5, 5)
(6, 5)
(1, 6)
(2, 6)
(3, 6)
(4, 6)
(5, 6)
(6, 6)
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
Write down the sample space for the following experiments:
1. Simultaneously tossing four coins.
2. Simultaneously rolling 4 standard six sided dice.
3. (E. Kreyszig: Advanced Engineering Mathematics, Problem set 24.2, pp1017)
Tossing a coin until the first Head appears.
4. (E. Kreyszig: Advanced Engineering Mathematics, Problem set 24.2, pp1017)
Rolling a die until the first Six appears.
5. (E. Kreyszig: Advanced Engineering Mathematics, Problem set 24.2, pp1018)
Choosing a committee of 2 from a group of 5 people.
6. Drawing a card from a standard deck of 52 cards.
7. Recording the lifetime of each of 3 light bulbs.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Event
An event is a subset of the sample space. In other words it is a
collection of outcomes.
In practice events are sometimes explicitly written out explicitly as
a set, while at other times it is described in sentences. To work out
problems we will often need to convert events expressed in
sentences to sets.
Key words to look out for when trying to understand how to write
an event as a set: and, or, not.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Rolling a standard six sided die.
The sample space can be written as:
S = {1, 2, 3, 4, 5, 6}
Let E be the event that the outcome is an odd number. We can
write:
E = {1, 3, 5}
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Simultaneously rolling two dice.
The sample space can be written as:
S = {(i, j) : i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3, 4, 5, 6}
Let E be the event that at least one of the dice shows 4. We can
write:
E = {(4, 1), (4, 2),(4, 3), (4, 4), (4, 5), (4, 6),
(1, 4), (2, 4), (3, 4), (5, 4), (6, 4)}
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Unions, Intersections and Complements
Recall, that for events (subsets) A and B of the sample space S:
The union A ∪ B of A and B consists of all points in A or B
or both.
The intersection A ∩ B of A and B consists of all points that
are in both A and B. We say that the A and B are mutually
exclusive if A ∩ B = ∅. A sequence of events E1 , E2 , E3 , . . .
are called mutually exclusive if Ei ∩ Ej = ∅ ∀ i 6= j.
The complement Ac of A is set of all points of S not in A.
The set difference A \ B is the set of all points in A but not
in B.
Note: Venn Diagrams offer a clear visual understanding of the
above processes.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Rolling a standard six sided dice.
S = {1, 2, 3, 4, 5, 6}
Let A be the event that the outcome is greater that 3, and B be
the event that the outcome is odd.
Then A ∩ B = {4, 5, 6} ∩ {1, 3, 5} = {5} is the event that the
outcome is odd and greater that 3.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Simultaneously rolling two dice.
S = {(i, j) : i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3, 4, 5, 6}
Let A be the event that at least one of the dice shows a 3, and B
be the event that the sum of the numbers on the two dice is 4.
Then
A ∪ B ={(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3),
(3, 1), (3, 2), (3, 4), (3, 5), (3, 6)} ∪ {(1, 3), (2, 2), (3, 1)}
={(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3),
(3, 1), (3, 2), (3, 4), (3, 5), (3, 6), (2, 2)}
is the event that at least one of the dice shows a 3 or the sum of
the two dice is 4.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. Three coins are tossed simultaneously. Write down the event
“at least one head appears” as a subset of the sample space.
2. Three coins are tossed simultaneously. Write down the event
“at most one head appears” as a subset of the sample space.
3. Three coins are tossed simultaneously. Write down the event
“exactly one head appears” as a subset of the sample space.
4. Three dice are rolled simultaneously. Write down the event
“the sum is at most 6” as a subset of the sample space.
5. Using Venn Diagrams show that A \ B = A ∩ (B c ).
6. Using Venn Diagrams graph and check the De Morgan’s
laws
(A ∪ B)c = Ac ∩ B c
(A ∩ B)c = Ac ∪ B c
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Section 3
Axiomatic approach to probability
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Warm-up definition of probability
Suppose the sample space S is finite that are equally likely, then
the probability P(A) of an event A can be defined as
P(A) =
#A
#S
where # denoted the cardinality of the set.
In particular it follows that
P(S) =
#S
=1
#S
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Tossing a fair coin.
S = {H, T }
1
#{H}
=
#{H, T }
2
#{T }
1
P({T }) =
=
#{H, T }
2
P({H}) =
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Simultaneously tossing a pair of unbiased coins.
S = {HH, HT , TH, TT }
Let E be the event that at least one of the coins shows heads. We
can write
E = {HH, HT , TH}
and probability that this event occurs:
P(E ) =
#E
3
=
#S
4
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Probability as a relative frequency
The warm-up definition has two clear drawbacks:
We are assuming that the sample space is finite.
We are assuming that the outcomes are equally likely.
In real life the above two assumptions may not be true.
Suppose the outcomes are not equally likely. We could conduct
several trials of the experiment and check in how many of those
trial the outcome belongs to the event.
The relative frequency of E = frel (E ) =
Number of times E occurs
Number of trials
Let n be the number of trials and n(E ) = Number of times E
occurs. Then the relative frequency of E can be written as
frel (E ) =
n(E )
n
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Inspecting a light-bulb.
S = {Defective, Okay}
We test 1000 light-bulbs and find out that 16 of them are defective
and the rest are Okay.
Then the relative frequency
16
= 0.016
1000
984
frel ({Okay }) =
= 0.984
1000
frel ({Defective}) =
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Tossing a biased coin, i.e. heads and tails are not
equally likely.
We toss the coin 500 times and find out that we get tails a total of
225 times.
Then the relative frequency
275
= 0.55
500
225
frel ({T }) =
= 0.45
500
frel ({H}) =
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Measuring lifetime of a micro-organism.
Suppose we observe 2000 individuals of a certain species of a
micro-organism and observe that 1543 of them lives for less than
24 hours.
Then
1543
2000
= 0.715
frel ({Micro-organism lives for less than 24 hours}) =
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Probability as a relative frequency
So at this point of time, we are tempted to define the probability
of the event E to be
P(E ) = frel (E ) =
n(E )
Number of times E occurs
=
Number of trials
n
However unfortunately this definition is dependent on the number
of trials.
We could try and rectify this problem by taking the limit
P(E ) = lim
n→∞
n(E )
n
However, we still have a drawback, its not clear if the limit exists!
We could circumvent this drawback by assuming that this limit
exists but such a truth is not very ‘self-evident’.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Lets check the dictionary
We get the following definition from dictionary.com
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
An axiomatic approach to define probability
Motivated from our earlier attempts, could we establish a list of
self-evident or acceptable truths which we use to create a
definition of probability?
We assume that given a sample space S, for every event E ⊆ S,
we can associate a value P(E ), referred to as the probability of
E , and this value satisfies three axioms known as the axioms of
probability.
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Three axioms of probability
Axiom 1:
0 ≤ P(E ) ≤ 1
Axiom 2:
P(S) = 1
Axiom 3:
For any sequence of mutually exclusive events E1 , E2 , E3 , . . . (i.e.
Ei ∩ Ej = ∅ ∀i 6= j),
!
∞
∞
[
X
P
Ei =
P(Ei )
i=1
i=1
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Consistency check
It turns out that the warm-up definition of probability, where we
have a finite sample space with outcomes being equally likely is
consistent with the axiomatic definition.
P(E ) =
#E
#S
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Basic theorems of probability: Complement rule
Theorem (Complement rule)
For an event E and its complement E c in a sample space S,
P(E c ) = 1 − P(E )
Proof:
S = E ∪ E c and E ∩ E c = ∅. Hence an application of Axiom 2
followed by Axiom 3 gives us:
1 = P(S) = P(E ∪ E c ) = P(E ) + P(E c ) =⇒ P(E c ) = 1 − P(E )
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Three unbiased coins are tossed simultaneously.
What is the probability that at least one head turns up?
Here the sample space consists of 8 sample points. Indeed, for the
sample space can be written as
S = {HHH, HHT , HTH, HTT , THH, THT , TTH, TTT }
The event that at least one head turns up is
E = {HHH, HHT , HTH, HTT , THH, THT , TTH}
and hence the required probability is
P(E ) =
7
#E
=
#S
8
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Example
Experiment: Three unbiased coins are tossed simultaneously.
What is the probability that at least one head turns up?
Instead of directly computing the probability as in the last slide, we
can observe that the complement of the given event can be
described as “no heads shows up”
E c = {TTT }
and the probability of this event is
P(E c ) =
#E c
1
=
#S
8
We use the complement rule to obtained the required probability,
P(E ) = 1 − P(E c ) = 1 −
1
7
=
8
8
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. Five unbiased coins are tossed simultaneously. What is the
probability that at least one heads shows up?
2. Five unbiased coins are tossed simultaneously. What is the
probability that at least two heads shows up?
3. (E. Kreyszig: Advanced Engineering Mathematics, Problem set 24.3, pp1024)
Three fair dice are rolled simultaneously. What is the
probability of obtaining a sum not greater than 16?
4. (E. Kreyszig: Advanced Engineering Mathematics, Problem set 24.3, pp1024)
Two fair dice are rolled simultaneously. What is the
probability of a sum greater than 3 but not exceeding 6?
5. A card is randomly drawn from a standard deck of 52 cards.
What is the probability that the card is not an Ace?
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Addition rule for arbitrary events
Theorem (Addition rule for arbitrary events)
For events E1 and E2 in a sample space S,
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Proof:
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Addition rule for arbitrary events
Using Axiom 3, we get:
A = (A ∩ B c ) t (A ∩ B)
B = (A ∩ B) t (Ac ∩ B)
=⇒ P(A) = P(A ∩ B c ) + P(A ∩ B)
=⇒ P(B) = P(A ∩ B) + P(Ac ∩ B)
=⇒ P(A ∩ B c ) = P(A) − P(A ∩ B)
=⇒ P(Ac ∩ B) = P(B) − P(A ∩ B)
Again using Axiom 3 followed by the above two equations, we get:
A ∪ B = (A ∩ B c ) t (A ∩ B) t (Ac ∩ B)
=⇒ P(A ∪ B) = P(A ∩ B c ) + P(A ∩ B) + P(Ac ∩ B)
=⇒ P(A ∪ B) = P(A) − P(A ∩ B) + P(A ∩ B) + P(B) − P(A ∩ B)
=⇒ P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Examples
(E. Kreyszig: Advanced Engineering Mathematics, Example 4, pp1022)
In tossing a fair die, what is the probability of getting an odd
number or a number less than 4?
S = {1, 2, 3, 4, 5, 6}
A = Getting an odd number = {1, 3, 5}
P(A) = #A/#S = 3/6
B = Getting a number less than 4 = {1, 2, 3}
P(B) = #B/#S = 3/6
A ∩ B = Getting an odd number less than 4 = {1, 3}
P(A ∩ B) = #A ∩ B/#S = 2/6
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
= 3/6 + 3/6 − 2/6 = 2/3
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
Exercises
1. Prove the following statements:
1 P(∅) = 0.
2 If E ⊆ F ⊆ S, then P(E ) ≤ P(F ) ≤ 1.
3 P(A ∪ B ∪ C ) = P(A) + P(B) + P(C ) − P(A ∩ B) − P(B ∩
C ) − P(C ∩ A) + P(A ∩ B ∩ C ).
2. Prove or Disprove:
1 If E ( F , then P(E )
P(F ).
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What is probability theory?
Experiments, Outcomes, Events
Axiomatic approach to probability
End of chapter
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