2. Probability is the likelihood or chance that
a particular event will or will not occur;
The theory of probability provides a
quantitative measure of uncertainty of
occurrence of different events resulting
from a random experiment, in terms of
quantitative measures ranging from 0 to 1;
3. Experiment: it is a process which produces
outcomes; Example, tossing a coin is an
experiment; an interview to gauge the job
satisfaction levels of the employees in an
organization is an experiment;
Event: it is the outcome of an experiment;
Example, if the experiment is to toss a fair
coin, an event can be obtaining a head; if an
event has a single possible outcome, then it
is a simple (or elementary) event; a subset of
outcomes corresponding to a specific event is
called an event space.
4. Independent & Dependent Events: two
events are said to be independent, if the
occurrence or non-occurrence of one is not
affected by the occurrence or non-occurrence
of the other; vice versa
Mutually Exclusive Events: two or more
events are said to be mutually exclusive if the
occurrence of one implies that the other
cannot occur; if X and Y are mutually
exclusive, then P(X∏Y)=0
Sample Space: denoted by S; it is the set of
all possible outcomes in an experiment;
6. This approach happens to be the earliest;
This school of thought assumes that all the possible
outcomes of an experiment are mutually exclusive &
equally likely;
If there are ‘a’ possible outcomes favorable to the
occurrence of Event E, & ‘b’ possible outcomes
unfavorable to the occurrence of Event E & all these
possible outcomes are equally likely & mutually
exclusive, then the probability that the event E will
occur, denoted by P(E), is
P(E)= Number of outcomes favorable to occurrence of E
Total number of outcomes
7. This approach has two characteristics:
a. The subjects refers to fair coins, fair
decks of cards; but if the coin is
unbalanced or there is a loaded dice, this
approach would offer nothing but
confusion;
b. In order to determine probabilities, no
coins had to be tossed, no cards
shuffled, i.e. no experimental data were
required to be collected;
8. This method uses the relative frequencies of
past occurrences as the basis of computing
present probability; hence it is based on
experiments conducted in the past;
If an Event ‘E’ has occurred ‘r’ number of
times in a series of ‘n’ independent trials;, all
under uniform conditions, then the ratio of
‘r’ gives the probability of Event ‘E’ provided
‘n’ is sufficiently large:
P(E)= r = favorable trials
n total of trials
9. This approach is based on the intuition of
an individual;
This is not a scientific approach;
It is based on accumulation of
knowledge, understanding and experience
of an individual;
10. For any event probability lies between 0 &
1;
It is represented in
percentages, ratios, fractions;
Each event has a complementary event
i.e. P(E1) + P’(E1) =1
12. It is the first type of probability;
A marginal or unconditional probability is
the simple probability of the occurrence of
an event;
Denoted by P(E) where ‘E’ is some event;
P(E)= Number of outcomes favorable to occurrence of E
Total number of outcomes
13. Second type of probability;
If E1 & E2 are two Events, then Union
probability is denoted by P(E1 U E2 );
It is the probability that Event E1 will occur or
that Event E2 will occur or both Event E1 &
Event E2 will occur;
For example, union probability is the
probability that a person either owns a Maruti
800 or Maruti Zen. For qualifying to be part of
the union, a person has to have atleast one of
these cars
14. It is the third type of probability;
If E1 & E2 are two Events, then Joint
probability is denoted by P(E1∏E2 );
It is the probability of the occurrence of
Event E1 and Event E2;
For example, it is the probability that a
persons owns both a Maruti 800 & Maruti
Zen; for joint probability, owning a single
car is not sufficient;
15. It is the fourth type of probability;
Conditional Probability of two Events E1 &
E2 is generally denoted by P(E1/E2);
It is probability of the occurrence of E1
given that E2 has already occurred;
Conditional probability is the probability
that a person owns a Maruti 800 given that
he already has a Maruti Zen;
16. Used to estimate union probability;
If there are two Events E1 & E2, then the
general rule of addition is given by:
P(E1 or E2) = P(E1) + P(E2) – P (E1 & E2);
P(E1 U E2) = P(E1) + P(E2) – P (E1∏E2);
Special Rule of addition for mutually
exclusive:
P(E1 or E2) = P(E1) + P(E2);
P(E1 U E2) = P(E1) + P(E2);
17. Used to estimate joint probability and also
conditional probability;
If there are two Events E1 & E2, then the
general rule of multiplication is given by:
P(E1 & E2) = P(E1) . P(E2 /E1);
P(E1 ∏ E2) = P(E1) . P(E2 /E1) ;
Special Rule of multiplication for
independent events:
P(E1 & E2) = P(E1) . P(E2);
P(E1 ∏ E2) = P(E1) . P(E2);
18. Bayes’ theorem was developed by Thomas
Bayes. In fact, Bayes’ theorem is an
extended use of the concept of conditional
probability;
The law of conditional probability is given
by:
P(E1/E2) = P(E1 ∏ E2) = P(E1) . P(E2 /E1)
P(E2) P(E2)
19.
20. A random variable is a variable which
contains the outcome of a chance
experiment; for example, in an experiment to
measure the number of customers who arrive
in a shop during a time interval of 2 minutes;
the possible outcome may vary from 0 to n
customers; these outcomes (0,1,2,3,4,…n)are
the values of the random variable.
These random variables are called discrete
random variables
21. In other words , a random variable which assumes
either a finite number of values or a countable infinite
number of possible values is termed as Discrete
Random variable
On the other hand, random variables that assumes any
numerical value in an interval or can take values at
every point in a given interval is called continuous
random variable. For example, temperatures recorded
for a particular city can assume any number like 32O
F, 32.5O F 35.8O F
Experiment outcomes which are based on
measurement scale such as time, distance, weight &
temperature can be explained by Continuous Random
variable
22. Most commonly used & widely known distribution
among all discrete distributions.
It is a sequence of repeated trials, called Bernoulli
Process which is characterized by:
1. Only two mutually exclusive outcomes are
possible;( one is referred to as success & the
other as failure)
2. The outcomes in a series of trials/observation
constitute independent events;
3. Probability of success (p) or failure (q) is constant
over a number of trials;
4. The number of events is discrete & can be
represented by integers(0,1,2,3,4,onwards)
23. P(X)= nCxpxqn-x
where
n= total number of trials
x = Designated value
p= probability of success
q= probability of failure
nCx= n!___
x!(n-x)!
24. It is named after the famous French
Mathematician Simeon Poisson;
It is also a discrete distribution; but there are
a few differences between Binomial &
Poisson distributions. For a given number of
trials the binomial distribution describes a
distribution of two possible outcomes: either
success or failure whereas Poisson focuses
on the number of discrete occurrences over
an interval.
It is widely used in the field of managerial
decision making; widely used in queuing
models
25. The event occur in a continuum of time &
at a randomly selected point & event either
occurs or doesn’t occur;
Whether the event occur or doesn’t occur
at a point, it is independent of the previous
point where the event may have occurred
or not;
The probability of occurrence of events
remains same/constant over the whole
period or throughout the continuum;
26. P(x/)= x e-
x!
(greek letter lambda) =mean/average
e (constant)= 2.71826
x is a random variable(designated
number)
27. It is the most commonly used distribution
among all probability distributions;
It has a wide range of practical application
example, where the random variables are
human characteristics such as height,
weight, speed, IQ scores;
Normal distribution was invented in the
18th century;
28. The curve of normal distribution is symmetrical/
mesokurtic;
The mean, median & mode are identical;
The two tail of normal curve asymptotic;
Curve is unimodal;
The total area under normal distribution is 100% &
the distribution is as follows:
µ+1σ = 68%
µ+2σ =97%
µ+3σ = 99.7%
Z= x- µ
σ