Probability Theory | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern

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 AꓵB.

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 (AꓵB) = 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 ꓵB).

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 ꓵB) = P (A) x P (B).

The theorem can be extended to three or more independent events. Thus: P (AꓵBꓵC) = 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 ꓵB)/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 ꓵB)/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 X1, X2, X3, …….,Xk, with respective probabilities P1, P2, P3…….P4 when P1 + P2 + P3 + …….., Pk = 1 the mathematical expectation of X denoted by E (X) is defined as: E (X) = p1X1 + P2X2 + P3X3 + …….+ P4X4.

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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