Fundamentals of Probability: Understanding Random Variables and Probability Distributions"

Probability
Probability is a branch of mathematics that deals with quantifying uncertainty. It measures how likely an event is to occur, expressed as a number between 0 and 1. A probability of 0 means the event will not occur, while a probability of 1 means the event is certain to occur. Probabilities are often expressed as fractions, decimals, or percentages. The foundational rules of probability include the sum of the probabilities of all possible outcomes of a random experiment equaling 1, and the probabilities of mutually exclusive events adding up.
Random Variable
A random variable is a variable that takes on numerical values determined by the outcome of a random phenomenon. It essentially assigns numbers to outcomes of a random process. There are two main types of random variables:

Discrete Random Variable: Takes on a countable number of distinct values, such as the result of rolling a die (1, 2, 3, 4, 5, or 6).
Continuous Random Variable: Takes on an uncountable number of possible values, such as the exact height of individuals in a population.
Probability Distribution
A probability distribution describes how the probabilities are distributed over the values of the random variable. It provides a function that maps each value of the random variable to its corresponding probability. The two main types of probability distributions are:

Discrete Probability Distribution: Applies to discrete random variables and is often represented by a probability mass function (PMF). For example, the PMF for rolling a fair six-sided die assigns a probability of
1
6
6
1
​
to each of the outcomes 1 through 6.

Continuous Probability Distribution: Applies to continuous random variables and is represented by a probability density function (PDF). The area under the PDF curve within a given range represents the probability that the random variable falls within that range. For example, the normal distribution, a common continuous distribution, is defined by its mean and standard deviation, and its PDF has the familiar bell-shaped curve.
Rolling a Die
Consider rolling a fair six-sided die:

Probability: The probability of rolling any specific number (e.g., 4) is
1
6
6
1
​
.
Random Variable: Let
𝑋
X be the random variable representing the outcome of the roll, where
𝑋
X can be 1, 2, 3, 4, 5, or 6.
Probability Distribution: The PMF of
𝑋
X is:
𝑃
(
𝑋
=
𝑥
)
=
{
1
6
if 
𝑥
∈
{
1
,
2
,
3
,
4
,
5
,
6
}
0
otherwise
P(X=x)={
6
1
​

0
​

if x∈{1,2,3,4,5,6}
otherwise
​
These concepts form the basis for more complex topics in probability and statistics, including expectation, variance, and stochastic processes.

1. 1Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
UNIT-I
PROBABILITY CONCEPTS
RANDOM VARIABLES
PROBABILITY DISTRIBUTION

2. 2Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Uncertainties
Managers often base their decisions on an analysis
of uncertainties such as the following:
What are the chancesthat sales will decrease
if we increase prices?
What is the likelihooda new assembly method
method will increase productivity?
What are the oddsthat a new investment will
be profitable?

3. 3Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Probability
Probabilityis a numerical measure of the likelihood
that an event will occur.
Probability values are always assigned on a scale
from 0 to 1.
A probability near zero indicates an event is quite
unlikely to occur.
A probability near one indicates an event is almost
certain to occur.

4. 4Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Probability as a Numerical Measure
of the Likelihood of Occurrence
0 1.5
Increasing Likelihood of Occurrence
Probability:
The event
is very
unlikely
to occur.
The occurrence
of the event is
just as likely as
it is unlikely.
The event
is almost
certain
to occur.

5. 5Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Statistical Experiments
In statistics, the notion of an experiment differs
somewhat from that of an experiment in the
physical sciences.
In statistical experiments, probability determines
outcomes.
Even though the experiment is repeated in exactly
the same way, an entirely different outcome may
occur.
For this reason, statistical experiments are some-
times called random experiments.

6. 6Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
An Experiment and Its Sample Space
An experimentis any process that generates well-
defined outcomes.
The sample spacefor an experiment is the set of
all experimental outcomes.
An experimental outcome is also called a sample
point.

7. 7Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
An Experiment and Its Sample Space
Experiment
Toss a coin
Inspection a part
Conduct a sales call
Roll a die
Play a football game
Experiment Outcomes
Head, tail
Defective, non-defective
Purchase, no purchase
1, 2, 3, 4, 5, 6
Win, lose, tie

8. 8Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Types of Experiments
Definite Experiment
Iftheoutcomesoftrials
ofanexperimentareunique
ordefinitethenthe
experimentiscalleddefinite
Experiment.
Ex:Everytimeaballcomes
toearthwhenitisthrown
towardssky.
Random Experiment
In an experiment if the outcome
of each trial is not unique but,
Having one of the several possible
Outcomes such as experiment is
Called random experiment.
Ex: The experiment of tossing a
Coin.

9. 9Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Mathematical Definition of Probability
Probability,itissaid,istheratioofthenumberof
favorablecasestothetotalnumberofequallylikely
cases.Probabilityofoccurrenceof‘A’isdenotedby
P(A).
P(A) = M/N
Where M= favorable cases
N= total no of equally likely cases.
Chance of occurrence of an event is called
probability

10. 10Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Ex: (1). If a coin is tossed, there are two equally likely
results, a head or a tail, hence
Probability of Head P(A) = 1/2
Ex:(2).Ifadiceisthrown,theprobabilityof
obtaininganevennumberisP(A)=3/6=1/2

11. 11Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Ex:(3).Fromabagcontaining10blackand20white
balls,aballisdrawnatrandom.Whatisthe
probabilitythatitisblack?
Ans: Total number of balls = 10+20 = 30
Total number of black balls = 10
Total number of white balls = 20
Probability of getting a black ball from a bag is
P(A) = 10/30 = 1/3
Probability of not getting a black ball P(B) = 1-P(A)
=1 –1/3 = 2/3
Note: P(A)+P(B) = 1
P(B)=1-P(A)

12. 12Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Ex:(4).Whatistheprobabilityofgettingakingfrom
packofcards?
A)4/52B)5/52C)6/52
Ex: Two dice are thrown simultaneously. The
probability of getting a sum of 9 is:

13. 13Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer:
Total cases:
(1,1) (1,2) …………………(1,6)
(2,1) (2,2)…………………..(2,6)
(3,1) (3,2)…………………...(3,6)
(4,1) (4,2)……………………(4,6)
(5,1) (5,2)…………………….(5,6)
(6,1) (6,2)……………………..(6,6)
Total cases in which sum of 9 can be obtained are:
(5, 4), (4, 5), (6, 3), (3, 6)
The probability of getting sum of P (9) = 4/36 = 1/9

14. 14Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Assigning Probabilities
Basic Requirements for Assigning Probabilities
1. The probability assigned to each experimental
outcome must be between 0 and 1, inclusively.
0 <P(E
i) <1 for all i
where:
E
iis the ithexperimental outcome
and P(E
i) is its probability

15. 15Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Event: Outcome or results is called event.
Types of events:
1.Mutually exclusive events
2.Independent and Dependent events
3.Equally likely events
4.Exhaustive events
5.Complementary events

16. 16Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
1.Mutually exclusive events
Two events are said to be mutually exclusive or
incompatible when both caanothappen simultaneously in a
single trial.
Ex: if a single coin is tossed either head can be up or tail can
be up, both cannot be up at the same time.
2.Independent and Dependent events
Two or more events are said to be independent when the
outcome of one does not effect, and is not effected by the
other.
Ex: If a coin is tossed twice the result of the second throw
would in no way be affected by the result of the first throw.
Dependent event: Dependent events are those in which the
occuranceor non-occurenceof one event in any one trial
affect the probability of other events in other trials.

17. 17Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Ex: if a card is drawn from a pack of playing cards and is not
replaced, this will alter the probability that the second card
drawn is, say an ace, similarly, the probability of drawing a
queen from a pack of 52 cards is 4/52 or 1/13. but if the card
drawn(queen) is not replaced in the pack, the probability of
drawing again a queen is 3/51.
4. Equally likely events: Events are said to be equally likely
when one does not occur more often than the others.
5. Exhaustive events: events are said to be exhaustive when
their totality includes all the possible outcomes of a random
experiment.
6. Complementary events: let there be two events A and B.
A is called the complementary event of B(vice versa) if A
and B are mutually exclusive and exhaustive for example,
when a dice is thrown, occuranceof an even numbers
{2,4,6} and odd number {1,3,5} are complementary events.

18. 18Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Mutually Exclusive Events
If events Aand Bare mutually exclusive, P(AB= 0.
The addition law for mutually exclusive events is:
P(AB) = P(A) + P(B)
There is no need to
include “-P(AB”

19. 19Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Theorems of Probability
1.The Addition Theorem
2.The Multiplication

20. 20Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
It states that if two events A and B are mutually exclusive
the probability then the probability of the occurrence of
either A or B is the sum of the individual probability of
A & B
1.Addition Theorem
It is written as:
P(A or B) = P(A) + P(B)
or
P(A U B) = P(A) + P(B)
or
P(A U B U C) = P(A) + P(B) + P(C)

21. 21Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 1:
One card is drawn from a standard pack of 52. what is
the probability that it is either a king or queen.
Answer:
There are 4 kings and 4 queens in a pack of 52 cards
The probability that the card drawn is King
= 4 C 1 / 52 = 4/52 ( n C r = n! / (n-r) ! r ! )
and
The probability that card drawn is queen
= 4/52
Since the events are mutually exclusive, the probability that the card
drawn is either a king or a queen:
4/52 + 4/52 = 8/52 = 2/13

22. 22Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
One card is drawn from a standard pack of 52. what is
the probability that it is either a Spade or Diamond.

23. 23Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
One card is drawn from a standard pack of 52. what is
the probability that it is either a spade or diamond.
Answer:
There are 13 spades and 13 diamonds in a pack of 52 cards
The probability that the card drawn is spade
= 13 C 1 / 52 = 13/52 ( n C r = n! / (n-r) ! r ! )
& The probability that card drawn is Diamond
= 13/52
Since the events are mutually exclusive, the probability that the card
drawn is either a spade or diamond :
13/52 + 13/52 = 26/52 = 1/2

24. 24Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
7. Not mutually exclusive events
When events are not mutually exclusive or it is possible for
both events to occur, the addition rule must be modified.
Ex: what is the probability of drawing either a king or heart
from a standard pack of cards?
Answer: it is obvious that the events king and heart can
occur together as we can draw a king or heart.
The probability of one or more of two events that are not
mutually exclusive we use the modified form of the addition
theorem.
P(A or B) = P(A) + P(B)-P(A∩B)
In case of three events:
P(AUBUC) = P(A)+P(B)+P(C)-P(A∩B)-P(A∩C)-P(B∩C)+
P(A∩B∩C)

25. 25Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 1:
What is the probability of drawing a king or a heart
Where the events are not mutually exclusive
events?
Answer:
P(king or heart) = P(king) + P(heart) –P(king & heart)
= 4/52 + 13/52 –1/52
= 4/13

26. 26Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
ThemanagingcommitteeofVaishaliwelfareassociation
formedasub-committeeof5personstolookinto
electricityproblem.Profilesofthe5personsare:
1.Male age: 40
2.Male age: 43
3.Female age 38
4.Female age 27
5.Male age 65
If a chairperson has to be selected from this, what is the
probability that person would be either female or over
30 years?

27. 27Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
ThemanagingcommitteeofVaishaliwelfareassociation
formedasub-committeeof5personstolookinto
electricityproblem.Profilesofthe5personsare:
1.Male age: 40
2.Male age: 43
3.Female age 38
4.Female age 27
5.Male age 65
If a chairperson has to be selected from this, what is the
probability that person would be either female or over
30 years?
Answer:
P(king or heart) = P(female) + P(over 30) –P(female &
over 30)
= 2/5 + 4/5 –1/5 = 1

28. 28Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Multiplication Theorem
If two events A and B are independent the probability
that they both will occureis equal to the theproduct
of their individual probability
P(A∩B) = P(A)xP(B)
If three events A, B & C are independent
P(A∩B ∩C) = P(A)xP(B)xP(C)

29. 29Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem1:
A man wants to marry a girl having qualities: white
complexion –the probability of getting such a girl is
one in twenty: handsome dowry –the probalityof
getting this is one in fifty: westernized manners –the
probability here is one in hundred.
Find out the probability of his getting
married to such a girl when the possession of these
three attributes is independent

30. 30Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Solution: probability of a girl with white complexion
=1/20 = 0.5
probability of a girl with handsome dowry
=1/50 = 0.02
probability of a girl with westernized manners
=1/100 = 0.01
The events are independent, the probability of
simultaneously occurrence of all thesqualities:
P(A∩B ∩C) = P(A)xP(B)xP(C)
= 0.05x0.02x0.01 = 0.0001

31. 31Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2: A problem in statistics is given to three
students A,B,C whose chances of solving it are 1/3, ¼
& 1/5 respectively. Find the probability that the
problem will be solved?

32. 32Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer:
P(A) = 1/3
P(B) = ¼
P(C) = 1/5
The probability of problem will be solved in
P(AUBUC) = P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-
P(A∩C)+P(A∩B∩C)
=1/3+1/4+1/5-1/3X1/4-1/4X1/5-
1/3X1/5+1/3X1/4X1/5 = 3/5 =0.6

33. 33Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Conditional Probability
IfA,Bareanytwodependenteventsthenthe
probabilityofhappeningevent‘B’giventhat‘A’has
alreadyhappenediscalledconditionalprobabilityofB/A
P(B/A) = P(A∩B)/P(A)
P(A∩B) = P(A)XP(B/A); IF P(A) ≠ 0.

34. 34Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Conditional Probability
Problem:1
Abagcontains5whiteand3blackballs.Twoballsare
Drawnatrandomoneaftertheotherwithoutreplacement.
Findtheprobabilitythatbothballsdrawnareblack.
probability of drawing a ball in the first attempt is
P(A) = 3/5+3 = 3/8
probability of drawing the 2
nd
ball given that
the first ball drawn is black(1
st
ball is not replaced)
P(B/A) = 2/5+2 = 2/7

35. 35Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Conditional Probability
Theprobabilitythatbothballsdrawnareblackisgivenby:
P(A∩B)=P(A)XP(B/A)=3/8X2/7=3/28
Problem2:iftwointegersareselected,atrandomand
Withoutreplacement,from{1,2,4,……99,100},whatisthe
Probabilitythatintegersareconsecutive?

36. 36Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Conditional Probability
Solution:giventhatSamplespace,S={1,2,3,4……99,100}
Twointegersareselectedatrandomin100C2ways
Withoutreplacement.
n(S)=4950
Let,Abeaneventofselectingtwoconsecutiveintegers.
A={(1,2)(2,3)(3,4)……(99,100)}
n(A)=99
Probabilityofselectingtwoconsecutiveintegersis,
P(A)=n(A)/n(S)
Substitutingthecorrespondingvaluesinaboveequation:
P(A)=99/4950=1/50

37. 37Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem1:Twodicearethrown.
Whatistheprobabilityofgetting
i.Thesumis10
ii.Atleast10.

38. 38Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.

39. 39Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.

40. 40Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.

41. 41Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
A manufacturer firm produces units of product
in four plants. Define event Ai: a unit is produced
in Plant i, i=1,2,3,4,5 and event B: a unit is defective,
from the past records of the propositions of defectives
Produced at each plant the following conditional
Probabilities are set:
P(B/A1) = 0.05
P(B/A2) = 0.10
P(B/A3) = 0.15
P(B/A4) = 0.02

42. 42Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 2:
Thefirstplantproduces30%oftheunitsofthe
product,Thesecondplant25%,thirdplant40%andthe
fourthplant5%,Aunitoftheproductmadeatoneof
Theseplantsistestedandisfoundtobedefective.
Whatistheprobabilitythattheunitwasproduced
inplant3?

43. 43Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Problem 3:
Acompanyhastwoplantstomanufacturescooters
Plant1-manufacture80%ofscooters
Plant2-manufacture20%ofscooters
Atplant185%outof100scootersareratedstandard
Qualityorbetter.Atpalnt2only65%outof100
scootersareratedstandardqualityorbelow:
i.Whatistheprobabilitythatscooterselectedat
randomcamefromplant1canbestandardqualityor
Better?
ii.Whatistheprobabilitythatscooterselectedat
randomcamefromplant2canbestandardqualityor
better?

44. 44Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.

45. 45Slide
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.