This document discusses probability theory and its applications, detailing key terminologies such as random experiments, sample spaces, and types of events including mutually exclusive and independent events. It explains various probability distributions including binomial, Poisson, and uniform distributions, as well as practical applications such as opinion polls and simulations. The document serves as a comprehensive introduction to the foundational concepts in probability theory and its relevance to statistical analysis.

1. Probability Queuing Theory
Presented By
M.PRAVEEN
S.SURYA
R.VENGATESH
M.YOGESH
S.VISHAL
GUIDED BY
Mr.GOPI(A.P.MATHS)

2. CONTENTS
 Terminologies
 Random variable
 Events
 Types of Events
 Probability distributions
 Example
 Practical applications

3. INTRODUCTION
 Probability theory is a very fascinating subject which
can be studied at various mathematical levels.
 Probability is the foundation of statistical theory and
applications.
 To understand probability , it is best to envision an
experiment for which the outcome (result) is unknown.
 Probability is the measure of how likely something will
occur.
 It is the ratio of desired outcomes to total outcomes.
(# desired) / (# total)

4. TERMINOLOGIES
 Random Experiment:
If an experiment or trial is repeated under
the same conditions for any number of times and it is
possible to count the total number of outcomes is called as
“Random Experiment”
 Sample Space:
The set of all possible outcomes of a
random experiment is known as “Sample Space” and
denoted by set S. [this is similar to Universal set in Set
Theory] The outcomes of the random experiment are
called sample points or outcomes.

5. Random variable
 Discrete Random Variable:
If the number of possible values of X
is finite or countably infinite then X is called a Discrete
Random Variable.
 Continuous Random Variable:
A random variable X is called a
Continuous Random Variable if X takes all possible values
in an interval.

6. Events
Definition:
An ‘event’ is an outcome of a trial meeting
a specified set of conditions other words, event is a subset
of the sample space S.
Events are usually denoted by capital
letters

7. Types of Events
 Exhaustive Events
 Favorable Events
 Mutually Exclusive Events
 Equally likely or Equi-probable Events
 Complementary Events
 Independent Events

8.  Exhaustive Events:
The total number of all possible elementary outcomes in a
random experiment is known as ‘exhaustive events’. In other
words, a set is said to be exhaustive, when no other possibilities
exists.
 Favorable Events:
The elementary outcomes which entail or favor the happening
of an event is known as ‘favorable events’ i.e., the outcomes which
help in the occurrence of that event.
 Mutually Exclusive Events:
Events are said to be ‘mutually exclusive’ if the occurrence of
an event totally prevents occurrence of all other events in a trial. In
other words, two events A and B cannot occur simultaneously.

9.  Equally likely or Equi-probable Events:
Outcomes are said to be ‘equally likely’
if there is no reason to expect one outcome to occur in
preference to another. i.e., among all exhaustive outcomes,
each of them has equal chance of occurrence.
 Complementary Events:
Let E denote occurrence of event. The
complement of E denotes the non occurrence of event E.
Complement of E is denoted by ‘Ē’.
 Independent Events:
Two or more events are said to be
‘independent’, in a series of a trials if the outcome of one
event is does not affect the outcome of the other event or
vise versa.

10. Two or more events
 If there are two or more events, you need
to consider if it is happening at the same time or one
after the other.
 “And”
If the two events are happening at the same
time, you need to multiply the two probabilities together.
 “Or”
If the two events are happening one after the
other, you need to add the two probabilities.

11. Probability distribution:
 Binomial Distribution:
A random variable ‘x’ is said to follow
binomial distribution if it assumes only non negative
values and its probability mass function is given by
p(X=x) = 𝑛𝑐 𝑥 𝑝 𝑥
𝑞 𝑛−𝑥

12.  Poisson Distribution:
A random variable ‘X’ taking non-
negative values is said to follow poisson distribution if its
probability mass function is given by

13.  Geometric Distribution:
A random variable ‘x’ is said to
follow geometric distribution if it assumes non-
negative values and its probability mass function
is given by
P(X=x) = 𝑞 𝑥
p ,where x=0,1,2,3……
where
p+q=1 , then q=1-p;
0 ≤ p ≤ 1

14. Continuous Distribution:
 Uniform Distribution:
A random variable ‘X’ is said to
follow uniform or rectangular distribution over an
interval(a,b) if its p.d.f is given by

15.  Exponential Distribution:
A continuous Random Variable ‘X’ defined
in (0,∞) is said to follow an exponential
distribution with parameters λ if its p.d.f is given
by
f(x) = λ𝑒−λ𝑥
where λ > 0 and 0 ˂ x ˂ ∞

16.  Gamma Distribution:
A continuous random variable ‘X’
is said to follow Gamma Distribution with parameters λ if
its p.d.f is given by

17. Example
 If I flip a coin, what is the probability of getting heads?
What is the probability of getting tails?
 Answer:
P(heads) = 1/2
P(tails) = 1/2

18. Another example
 If I roll a number cube and flip a coin:
What is the probability I will get a heads and a 6?
What is the probability I will get a tails or a 3?
 Answers
P(heads and 6) = 1/2 x 1/6 =1/12
P(tails or a 5) = 1/2 + 1/6 = 8/12 = 2/3

19. Practical applications
 Probability in opinion poll:
The actual probability often applies to the
percentage of a large group. Suppose you know that 60
percent of the people in your community are Democrats,
30 percent are Republicans, and the remaining 10
percentage Independents or have another political
affiliation. If you randomly select one person from your
community, what’s the chance the person is a Democrat?
The chance is 60 percent. You can’t say that the person
is surely a Democrat because the chance is over 50
percent; the percentages just tell you that the person is
more likely to be a Democrat. Of course, after you ask
the person, he or she is either a Democrat or not; you
can’t be 60-percent Democrat.

20.  Relative Frequency:
The approach is based on collecting data and, based on
that data, finding the percentage of time that an event
occurred. The percentage you find is the relative frequency
of that event — the number of times the event occurred
divided by the total number of observations made.
If you count 100 bird visits, and 27 of the visitors are
cardinals, you can say that for the period of time you
observe, 27 out of 100 visits or 27 percent, the relative
frequency — were made by cardinals. Now, if you have to
guess the probability that the next bird to visit is a
cardinal, 27 percent would be your best guess. You come
up with a probability based on relative frequency

21.  Simulation:
 Simulation approach is a process that creates data
by setting up a certain scenario, playing out that
scenario over and over many times, and looking at
the percentage of times a certain outcome occurs.
 It’s different in three ways:
You create the data (usually with a computer); you
don’t collect it out in the real world.
The amount of data is typically much larger than
the amount you could observe in real life.
You use a certain model that scientists come up
with, and models have assumptions.

22.  Statistics:
In statistics there is usually a collection of
random variables from which we make an observation and
then do something with the observation. The most common
situation is when the collection of random variables of
interest are mutually independent and with the same
distribution. Such a collection is called a random sample.
A statistic is a function of a random
sample that does not contain any unknown parameters.

23. THANK YOU

24. ANY QUERIES