**BUSINESS STATISTICS NOTES**

B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)

**Probability and Probability
Distributions**

**Part A:
Probability Theory**

**What is Probability? Explain with example.**

In
our day to day life, we came across situations in which we cannot predict with
absolute certainly about the exact occurrence of any future events such as
prediction of demand of a product, chances of rain, chance of winning a cricket
match by a particular team etc. In such situations we can guess or speculate
about the likelihood of occurrence say 50% or 90%. The likelihood of occurrence
of an event is known as Probability.

In
layman’s terminology the word probability connotes that there is uncertainty
about the happening of event. But in mathematics and statistics probability
means expression of likelihood or chances of occurrence of an event. A
probability is a number which ranges from 0 to 1 – zero for an event which
cannot occur and 1 for an event certain to occur.

For
example:

a) In a
cricket match, chances of winning a team are 50%.

b) If a coin
is tossed, chances of head are 50%.

c) If a dice
is thrown, chances of any one number are 16.67%.

The above
percentage is based on the classical definition of probability in which
probability is calculated by dividing total number of favourable case by total
number of possible outcomes.

**Why do we use probability? – Applications and
importance of Probability**

Initially
the applications of probability theories were restricted to games of chances.
But the passage of time they are used in taking various important business
decisions. Also probability theory is being applied in the solution of social,
economic, political and business problems. The insurance industry required
precise knowledge about the risk of loss in order to calculate premium. In
fact, probability has become an important part of our everyday lives. In
personal and management decisions, we face uncertainty and use probability
theory. It is not possible to forecast the future with 100 percent certainty in
any decision problem. The probability theory provides a tool to cope up with
uncertainty.

**What are four Types of Probability? – School
of thoughts of probability**

There
are four different ways of calculating or computing probability:

a) Subjective probability

b)
Objective Probability: It is divided into two parts:

1. Classical
or priori probability

2. Empirical
or relative approach of probability.

c) Axiomatic approach of probability

**a)
Subjective Probability:** This approach was introduced in the year 1926
by Frank Ramsey. The subjective probability is defined as the probability
assigned to an event by an individual based on whatever evidence is available.
Hence such probabilities are dependent on personal judgement and experience. It
may be influenced by the personal belief, attitude and bias of the person
applying it.

**b)
Objective Probability:** Objective probability is divided into two
categories – Classical probability and Empirical Probability.

**1.
Classical or Priori Probability:** The Classical approach to probability
is the oldest and simplest method of calculating probability. This approach is
mainly applicable when the outcomes of a random experiment are “equally
likely”. Here, A random experiment is an experiment whose all possible outcomes
are known and which can be repeated under identical conditions but it is not
possible to predict the outcome of any trial. An Event means the possible
outcomes of a random experiment.

For example: Tossing a coin is a perfect example of Random
experiment. There are two possible outcomes – either head or tail appears but
exact prediction is not possible in tossing. Also all the events are equally
likely.

Probability under classical approach is calculated with the help
of following formula:

P(A) = Total number of equally likely events / Total number of
possible outcomes.

In the above example, there are two possible outcomes head or tail
and chances of head is 1 out of 2 and also chance of tail is 1 out of 2. If
probability or chances of head is to be predicted, the

Required probability (Head) will be = ½.

**Limitations
of classical approach:**

Though this approach is simple to understand, but it suffers from
the following limitations:

a) It is applicable only when the total numbers of events are
finite.

b) It is applicable only when all the events are equally likely.

c) It has limited
applications like tossing a coin, throwing a dice, drawing cards from a well
shuffled pack where the possible events are known.

**2.
Empirical or relative approach of probability:** Due to
the various limitations of classical definitions of probability, there are
cases where we can use the statistical definition of probability based on the
concept of relative frequency. This definition of probability was first
developed by the British Mathematics in 1800s for calculating the risk of
losses in life insurance and commercial insurance. The empirical or relative is
based on the relative frequency of occurrence of an event of a large number of
repeated trials. When this approach is made, the following important points
should be noted:

a) The
probability so determined is only an estimate of the actual value.

b) The larger
the number of trials, the better the estimate of the probability.

c) The trials
should be conducted under identical conditions.

**c)
Axiomatic or set approach of probability:** This is the best approach of
calculating probability which was introduced in 1933 by Russian Mathematician
A.N. Kolmogorov. He introduces probability as a set function and is considered
as a classic. When this approach is followed, no precise definition of
probability is given, rather we are given certain axioms or postulates on which
probability calculations are based. The whole field of probability theory for
finite sample spaces is based upon the following three axioms:

1. The probability of an event ranges from 0 to 1 - zero for an
event which cannot occur and 1 for an event certain to occur.

2. The probability of the entire sample space is 1.

3. If two events A and B are mutually exclusive, then the
probability of occurrence of either A or B shall be:

P(AUB) = P(A) + P(B)

To have a clear idea about axiomatic approach of probability, we
must first of all understand set theory.

**Calculation of Probability**

At the time of calculation of probability in
any one the approach mentioned above, we must be familiar with the certain terms
which are explained below:

**Set:** A set is
a collection of items having some common features or characteristics. E.g. set
of all black car in our district.

**Sample
Space:** The set of all possible outcomes of an experiment is called the
sample space and it is denoted by S. E.g. set of even numbers of a dice S =
{2,4,6}. Each element of a sample space is called sample point. A sample space
may be finite or infinite.

**Union:** Union of
two events A and B is the set of all sample points belonging to A or B and is
denoted by: AUB.

**Intersection:** The
intersection of two events A and B is the set of all points common to both A
and B is denoted by Aê“µB.

**Trial: **In
statistics, trial means performance of a random experiment. For example:
tossing of a coin, throwing a dice, drawing a card from a well shuffled pack
etc.

**Random
experiment: **A random experiment is an experiment whose all possible outcomes
are known and which can be repeated under identical conditions but it is not
possible to predict the outcome of any trial. For example: Tossing a coin is a
perfect example of Random experiment. There are two possible outcomes – either
head or tail appears but exact prediction is not possible in tossing. Here the
chances of occurrence of events are equally likely.

**Events/Outcomes:
**Results
of a random experiment are called outcomes. Possible outcomes of a random
experiment are also called events.

**Equally Likely Events: **Events are
said to be equally likely when no one events can be expected to occur in
preference to the other events. For example, in tossing a coin head and tail
are equally likely events. All the cards of an well shuffled pack of cards are
equally likely when one card is drawn. Here, we can also say that probability
of event is same in case of equally likely items.

**Mutually Exclusive Events: **Two events
are said to be mutually exclusive or incompatible when both cannot happen
simultaneously in a single trial or, in other words, the occurrence of any one
of them make impossible the occurrence of the other. For example, if a single
coin is tossed either head can be up or tail can be up, both cannot be up at
the same time. To take another example, if we toss a die and observe 3, we
cannot expect 5 also in the same toss of die. Symbolically, if A and B are
mutually exclusive events, P (Aê“µB) = 0.

**Exhaustive Events: **Events are
said to be exhaustive when at least one of them must necessarily occur and
their totality includes all the possible outcomes of a random experiment. For
example, while tossing a die, the possible outcomes are 1, 2, 3, 4, 5 and 6 and
hence the exhaustive number of cases is 6.

**Complementary Events: **Let there
be two events A and B. A is called the complementary event of B (and vice
versa) if A and B are mutually exclusive and exhaustive. For example, when a
die is thrown, occurrence of an even number (2, 4, 6) and odd number (1, 3, 5)
are complementary events.

**Simple and Compound Events: **In case of
simple events we consider the probability of the happening or not happening of
single events. For example, we might be interested in finding out the
probability of drawing a red ball from a bag containing 10 white and 6 red
balls. On the other hand, in case of compound events we consider the joint
occurrence of two or more events. For example if a bag contains 10 white and 6
red balls if two successive drawn of 3 balls are made, we shall be finding out
the probability of getting 3 white balls in the first draw and 3 black balls in
the second draw – we are thus dealing with a compound event.

**Independent and Dependent Events: **Two or
more events are said to be independent when the outcome of one does not affect,
and is not affected by the other. For example, 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. Similarly, the results obtained by throwing a die are independent
of the results obtained by drawing an ace from a pack of cards.

Dependent
events are those in which the occurrence or non-occurrence of one event in any
one trial affects the probability of other events in other trials. For example,
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.

**Addition Theorem**

The
addition theorem states that if two events A and B are mutually exclusive the
probability of the occurrence of either A or B is the sum of the individual
probability of A and B.

Symbolically,
P (A or B) = P (A) + P (B)

But
when the events are not mutually exclusive, the addition rule must be modified
because there are some common elements in all the events. For example, what is
the probability of drawing either a king or spade if one card is drawn from a
well shuffled pack? It is obvious that king and spade can occur together as we
can draw a king of spade. We must reduce from the probability of drawing either
a king or a spade, the chance we can draw both of them together. Hence for
finding the probability of one or more of two events that are not mutually
exclusive we use the following addition theorem: P (A or B) = P (A) + P (B) - P
(A ê“µB).

**Multiplication Theorem**

This
theorem states that if two events A and B are independent, the probability that
they both will occur is equal to the product of their individual probabilities.
Symbolically, if A and B are independent, then:

P
(A ê“µB) = P (A) x P (B).

The
theorem can be extended to three or more independent events. Thus: P (Aê“µBê“µC) =
P(A)xP(B)x P(C).

**Conditional Probability**

The
multiplication theorem is applicable in case of independent events only. But in
case of dependent events, the concept of conditional probability is applicable.
Dependent events are those in which the occurrence or non-occurrence of one
event in any one trial affects the probability of other events in other trials.
Conditional probabilities of dependent events are calculated in two ways:

a)
If two events A and B are dependent, then the conditional probability of B if
it is given that A already happens:

P(B/A) =
P(A ê“µB)/P(A)

b)
If two events A and B are dependent, then the conditional probability of A if
it is given that B already happens:

P(A/B) =
P(A ê“µB)/P(B)

**Bayes Theorem**

Bayes’
theorem also known as Bayes’ rule was discovered in the year 1763 by Sir Thomas
Bayes. This theorem is used to determine the conditional probability of events.
The concept of conditional probability takes into account information about the
occurrence of one event to predict the probability of another event. Bayes’ theorem
is simply an extension of conditional probability. Bayes’ theorem is a revised
probabilities based on new information and to determine the probability that a
particular effect was due to a specific cause.

Bayes’
theorem is superior to classical theory because classical theory is mainly
empirical since it employs only sample information as the basis for estimation
and testing, while the Bayesian approach employs any and all available
information whether it is personal judgement or empirical evidence. Further, a
Bayesian inference can be made on prior information alone or on both prior and
sample information. Because of this Bayesian method can be considered as an
extension of the classical approach.

By using Bayes theorem, revised probability
can be found with the help of following formula:

P (A|B) = P (B|A).P(A)/P(B)

Here, P (A|B) is
the probability of A if event B already happens.

P (B|A) is the
probability of B if event A already happens.

P (A) represents
chances of event A.

P (B) represents
chances of event B.

**Mathematical Expectation**

The
concept of mathematical expectation is of great importance in statistical work.
The mathematical expectation (also called the expected value) of a random
variable is the weighted arithmetic mean of the variable, the weights used to
find the mathematical expectation are all the respective probabilities of the
values that the variable can possibly assume.

If
X denotes a discrete random variable which can assume the values X_{1},
X_{2}, X_{3}, …….,X_{k}, with respective probabilities
P_{1}, P_{2}, P_{3}…….P_{4} when P_{1}
+ P_{2} + P_{3} + …….., P_{k }= 1 the mathematical
expectation of X denoted by E (X) is defined as: E (X) = p_{1}X_{1}
+ P_{2}X_{2} + P_{3}X_{3} + …….+ P_{4}X_{4}.

Thus
the expected value equals the sum of each particular value within the set (X)
multiplied by the probability that X equals that particular value. It should be
noted that the concept of mathematical expectation was originally applied to
games of chance and lotteries, but the notion of an expected value has become
more generally applied and is now a common term in everyday parlance. Business
situations frequently involve the consideration of expected values.

**Properties of Mathematical
Expectations**

1) The
expected value of a constant is the constant itself i.e.,E(a) = a.

2) E(a+bx) =
a + bE (x), where a, b are constants.

3) If x and y
are random variables then: E(x __+__ y) = E(x) __+__ E(y).

4) If x and y
be two independent random variable then: E(xy) = E(x).E(y).

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