Author: indiafreenotes

The balance of payments (henceforth BOP) is a consolidated account of the receipts and payments from and to other countries arising out of all economic transactions during the course of a year.

In the words of C. P. Kindleberger: “The balance of payments of a country is a systematic record of all economic transactions between the residents of the reporting and the residents of the foreign countries during a given period of time.” Here by ‘residents’ we mean individuals, firms and government.

By all economic transactions we mean individuals, firms and government. By all economic transactions we mean transactions of both visible goods (merchandise) and invisible goods (services), assets, gifts, etc. In other words, the BOP shows how money is spent abroad (i.e., payments) and how money is received domestically (i.e., receipts).

Thus, a BOP account records all payments and receipts arising out of all economic transactions. All payments are regarded as debits (i.e., outflow of money) and are recorded in the accounts with a negative sign and all receipts are regarded as credits (i.e., inflow or money) and are recorded-in the accounts with a positive sign. The International Monetary Fund defines BOP as a “statistical statement that subsequently summarises, for a specific time period, the economic transactions of an economy with the rest of the world.”

Components of BOP Accounts

(A) The Current Account

The current account of BOP includes all transaction arising from trade in currently produced goods and services, from income accruing to capital by one country and invested in another and from unilateral transfers— both private and official. The current account is usually divided in three sub-divisions.

The first of these is called visible account or merchandise account or trade in goods account. This account records imports and exports of physical goods. The balance of visible exports and visible imports is called balance of visible trade or balance of merchandise trade [i.e., items 1(a), and 2(a) of Table 6.1].

The second part of the account is called the invisibles account since it records all exports and imports of services. The balance of these transactions is called the balance of invisible trade. As these transactions are not recorded—in the customs office unlike merchandise trade we call them invisible items.

It includes freights and fares of ships and planes, insurance and banking charges, foreign tours and education abroad, expenditures on foreign embassies, tran­sactions out of interest and dividends on foreigners’ investment and so on. Items 2(a) and 2(b) comprise services balance or balance of invisible trade in table 6.1.

The difference between merchandise trade and invisible trade (i.e., items 1 and 2) is known as the balance of trade.

There is another flow in the current account that consists of two items [3(a) and 3(b)]. Investment income consists of interest, profit and dividends on bonus and credits. Interest earned by a US resident from the TELCO share is one kind of investment income that represents a debit item here.

There may be a similar money inflow (i.e., credit item). Unrequited transfers include grants, gifts, pension, etc. These items are such that no reverse flow occurs. Or these are the items against which no quid pro quo is demanded. Residents of a country received these cost-free. Thus, unilateral transfers are one-way transactions. In other words, these items do not involve give and take unlike other items in the BOP account.

Thus the first three items of the BOP account are included in the current account. The current account is said to be favourable (or unfavourable) if receipts exceed (fall short of) payments.

(B) The Capital Account

The capital account shows transactions relating to the international movement of ownership of financial assets. It refers to cross-border movements in foreign assets like shares, property or direct acquisitions of companies’ bank loans, government securities, etc. In other words, capital account records export and import of capital from and to foreign countries.

The capital account is divided into two main subdivisions: short term and the long term move­ments of capital. A short term capital is one which matures in one year or less, such as bank accounts.

Long term capital is one whose maturity period is longer than a year, such as long term bonds or physical capital. Long term capital account is, again, of two categories: direct investment and portfolio investment. Direct investment refers to expenditure on fixed capital formation, while portfolio investment refers to the acquisition of financial assets like bonds, shares, etc. India’s investment (e.g., if an Indian acquires a new Coca- Cola plant in the USA) abroad represents an outflow of money. Similarly, if a foreigner acquires a new factory in India it will represent an inflow of funds.

Thus, through acquisition or sale and purchase of assets, capital movements take place. Investors then acquire controlling interests over the asset. Remember that exports and imports of equipment do not appear in the capital account. On the other hand, portfolio investment refers to changes in the holding of shares and bonds. Such investment is portfolio capital and the ownership of paper assets like shares does not ensure legal control over the firms.

[In this connection, the concepts of capital exports and capital imports require little elabo­ration. Suppose, a US company purchases a firm operating in India. This sort of foreign investment is called capital import rather than capital export. India acquires foreign currency after selling the firm to a US company. As a result, India acquires purchasing power abroad. That is why this transaction is included in the credit side of India’s BOP accounts. In the same way, if India invests in a foreign country,, it is a payment and will be recorded on the debit side. This is called capital export. Thus, India earns foreign currency by exporting goods and services and by importing capital. Similarly, India releases foreign currency by importing visible and invisibles and exporting capital.

(C) Statistical Discrepancy Errors and Omi­ssions

The sum of A and B (Table 6.1) is called the basic balance. Since BOP always balances in theory, all debits must be offset by all credits, and vice versa. In practice, it rarely happens—parti­cularly because statistics are incomplete as well as imperfect. That is why errors and omissions are considered so that the BOP accounts are kept in balance (Item C).

(D) The Official Reserve Account

The total of A, B, C, and D comprise the overall balance. The category of official reserve account covers the net amount of transactions by governments. This account covers purchases and sales of reserve assets (such as gold, convertible foreign exchange and special drawing rights) by the central monetary authority.

Now, we can summarise the BOP data

Current account balance + Capital account balance + Reserve balance = Balance of Payments

(X – M) + (CI – CO) + FOREX = BOP

X is exports,

M is imports,

CI is capital inflows,

CO is capital outflows,

FOREX is foreign exchange reserve balance.

BOP Always Balances

A nation’s BOP is a summary statement of all economic transactions between the residents of a country and the rest of the world during a given period of time. A BOP account is divided into current account and capital account. Former is made up of trade in goods (i.e., visible) and trade in services (i.e., invisibles) and unrequited transfers. Latter account is made up of transactions in financial assets. These two accounts comprise BOP

A BOP account is prepared according to the principle of double-entry book keeping. This accounting procedure gives rise to two entries— a debit and a corresponding credit. Any transaction giving rise to a receipt from the rest of the world is a credit item in the BOP account. Any transaction giving rise to a payment to the rest of the world is a debit item.

The left hand side of the BOP account shows the receipts of the country. Such receipts of external purchasing power arise from the commodity export, from the sale of invisible services, from the receipts of gift and grants from foreign govern­ments, international lending institutions and foreign individuals, from the borrowing of money from the foreigners or from repayment of loan by the foreigners.

The right hand side shows the payments made by the country on different items to the foreigners. It shows how the total of external purchasing power is used for acquiring imports of foreign goods and services as well as the purchase of foreign assets. This is the accounting procedure.

However, no country publishes BOP accounts in this format. Rather, by convention, the BOP figures are published in a single column with positive (credit) and negative (debit) signs. Since payments side of the account enumerates all the uses which are made up of the total foreign purchasing power acquired by this country in a given period, and since the receipts of the accounts enumerate all the sources from which foreign purchasing power is acquired by the same country in the same period, the two sides must balance. The entries in the account should, therefore, add up to zero.

In reality, why should they add up to zero? In practice, this is difficult to achieve where receipts equal payments. In reality, total receipts may diverge from total payments because of:

(i) The difficulty of collecting accurate trade information

(ii) The difference in the timing between the two sides of the balance

(iii) A change in the exchange rates, etc.

Because of such measurement problems, resource is made to ‘balancing item’ that intends to eliminate errors in measurement. The purpose of incorporating this item in the BOP account is to adjust the difference between the sums of the credit and the sums of the debit items in the BOP accounts so that they add up to zero by construc­tion. Hence the proposition ‘the BOP always balances’. It is a truism. It only suggests that the two sides of the accounts must always show the same total. It implies only an equality. In this book-keeping sense, BOP always balances.

Thus, by construction, BOP accounts do not matter. In fact, this is not so. The accounts have both economic and political implications. Mathe­matically, receipts equal payments but it need not balance in economic sense. This means that there cannot be disequilibrium in the BOP accounts.

A combined deficit in the current and capital accounts is the most unwanted macroeconomic goal of ,an economy. Again, a deficit in the current account is also undesirable. All these suggest that BOP is out of equilibrium. But can we know whether the BOP is in equilibrium or not? Tests are usually three in number:

(i) Movements in foreign exchange reserves including gold

(ii) Increase in borrowing from abroad

(iii) Movements in foreign exchange rates of the country’s currency in question.

Firstly, if foreign exchange reserves decline, a country’s BOP is considered to be in disequilibrium or in deficit. If foreign exchange reserves are allowed to deplete rapidly it may shatter the confidence of people over the domestic currency. This may ultimately lead to a run on the bank.

Secondly, to cover the deficit a country may borrow from abroad. Thus, such borrowing occurs when imports exceed exports. This involves payment of interest on borrowed funds at a high rate of interest.

Finally, the foreign exchange rate of a country’s currency may tumble when it suffers from BOP disequilibrium. A fall in the exchange rate of a currency is a sign of BOP disequilibrium.

Thus, the above (mechanical) equality between receipts and payments should not be interpreted to mean that a country never suffers from the BOP problems and the international economic transactions of a country are always in equilibrium.

Implications of an Unbalance in the BOP

Although a nation’s BOP always balances in the accounting sense, it need not balance in an economic sense.

An unbalance in the BOP account has the following implications:

In the case of a deficit

(i) Foreign exchange or foreign currency reserves decline,

(ii) Volume of international debt and its servicing mount up, and

(iii) The exchange rate experiences a downward pressure. It is, therefore, necessary to correct these imbalances.

BOP Adjustment Measures:

BOP adjustment measures are grouped into four:

(i) Protectionist measures by imposing customs duties and other restrictions, quotas on imports, etc., aim at restricting the flow of imports,

(ii) Demand management policies—these include restrictionary monetary and fiscal policies to control aggregate demand [C + I + G + (X – M)],

(iii) Supply-side policies—these policies aim at increasing the nation’s output through greater productivity and other efficiency measures, and, finally,

(iv) exchange rate management policies— these policies may involve a fixed exchange rate, or a flexible exchange rate or a managed exchange rate system.

As a method of connecting disequilibrium in a nation’s BOP account, we attach importance here to exchange rate management policy only.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people’s wages by reference to their respective education levels, where the individuals’ data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Additive Model:

1. Data is represented in terms of addition of seasonality, trend, cyclical and residual components
2. Used where change is measured in absolute quantity
3. Data is modeled as-is

Additive model is used when the variance of the time series doesn’t change over different values of the time series.

On the other hand, if the variance is higher when the time series is higher then it often means we should use a multiplicative models.

Returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1

If error’s increments have normal iid distributions then returni has also a normal distribution with constant variance over time.

Multiplicative model:

1. Data is represented in terms of multiplication of seasonality, trend, cyclical and residual components
2. Used where change is measured in percent (%) change
3. Data is modeled just as additive but after taking logarithm (with base as natural or base 10)

If log of the time series is an additive model then the original time series is a multiplicative model, because:

log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)

So the return of logarithms:

log(pricei)−log(pricei−1)=log(pricei/pricei−1)

A time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

Time series are very frequently plotted via line charts. Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called “time series analysis”, which focuses on comparing values of a single time series or multiple dependent time series at different points in time. Interrupted time series analysis is the analysis of interventions on a single time series.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people’s wages by reference to their respective education levels, where the individuals’ data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Analysis of time series has a lot of utilities for the various fields of human interest viz: business, economics, sociology, politics, administration etc. It is, also, found very useful in the fields of physical, and natural sciences. Some such points of its utilities are briefly here as under:

(i) It helps in studying the behaviours of a variable. 

In a time series, the past data relating to a variable over a period of time are arranged in an orderly manner. By simple observation of such a series, one can understand the nature of change that takes place with the variable in course of time. Further, by the technique of isolation applied to the series, one can tendency of the variable, seasonal change, cyclical change, and irregular or accidental change with the variable.

(ii) It helps in forecasting

The analysis of a time series reveals the mode of changes in the value of a variable in course of the times. This helps us in forecasting the future value of a variable after a certain period. Thus, with the help of such a series we can make our future plan relating to certain matters like production, sales, profits, etc. This is how in a planned economy all plans for the future development are the analysis of time series of the relevant data.

(iii) It helps in evaluating the performances.

Evaluation of the actual performances with reference to the predetermined targets is highly necessary to judge the efficiency, or otherwise in the progress of a certain work. For example, the achievements or out five-Year Plans are evaluated by determining the annual rate of growth in the gross national product. Similarly, our policy of controlling the inflation, and price rises is evaluated with the help of various price indices. All these are facilitated by analysis of the time series relating to the relevant variables.

(iv) It helps in making comparative study

Comparative study of data relating to two, of more periods, regions, or industries reveals a lot of valuable information which guide a management in taking the proper course of action for the future. A time series, per se, provides a scientific basis for making the comparision between the two, or more related set of data as in such series, the data are chronologically, and the effects of its various components are gradually isolated and unraveled.

Limitations

Forecasting

The central logical problem in forecasting is that the “cases” (that is, the time periods) which you use to make predictions never form a random sample from the same population as the time periods about which you make the predictions. This point is vividly illustrated by the 509-point plunge in the Dow-Jones Industrial Average on October 19, 1987. Even in percentage terms, no one-day drop in the previous 40 years (comprising some 8000 trading days) had ever been more than a fraction of that size. Thus this drop (which occurred in the absence of any dramatic news developments) could never have been forecast just from a time-series study of stock prices over the previous 40 years, no matter how detailed. It is now widely believed that a major cause of the drop was the newly-introduced widespread use of personal computers programmed to sell stock options whenever stock prices dropped slightly; this created a snowball effect. Thus the stock market’s history of the previous 40 or even 100 years was irrelevant to predicting its volatility in October 1987, because nearly all that history had occurred in the absence of this new factor.

The same general point arises in nearly all forecasting work. If you have records of monthly sales in a department store for the last 10 years, and are asked to project those sales into the future, those statistics will not reflect the fact that as you work, a new discount store is opening a few blocks away, or the city has just changed the street in front of your store to a one-way street, making it harder for customers to reach your store.

A second problem that arises in time-series forecasting is that you rarely know the true shape of the distribution with which you are working. Workers in other areas often assume normal distributions while knowing that assumption is not fully accurate. However, such a simplification may produce more serious errors in time-series work than in other areas. In much statistical work the problem of non-normal distributions is greatly ameliorated by the fact that you are really concerned with sample means, and the Central Limit Theorem asserts that means are often approximately normally distributed even when the underlying scores are not. However, in time-series forecasting you are often concerned with just one time period–the period for which you want to forecast. Thus the Central Limit Theorem has no chance to operate, and the assumption of normality may lead to seriously wrong conclusions. Even if your forecast is just one of a series of forecasts which you update after each new time period, the forecasts are made one at a time, so that a single seriously wrong forecast may bankrupt your company or lead to your dismissal, and nobody will ever learn that your next 50 forecasts would have been within the range predicted by a normal distribution. Some stock market speculators, who had previously been quite successful, were bankrupted or driven into retirement by the stock market plunge of 1987. That’s very different from the situation in which a company hires, all at once, 50 workers identified by a competence test. If one out of the 50 is a spectacular failure, the company (and you the forecaster) will survive because at the very same time the other 49 were turning out well.

Analyzing the Impact of Single Events

When you try to assess the impact of a single event, the major problem is that there are always many events occurring at any one time. Suppose you are trying to assess the effect of a new toll on bridge A on traffic across bridge B, but a new store opened near bridge B the same day the toll was introduced, permanently increasing traffic on bridge B. When critics remind us that “correlation does not imply causation”, they are mostly talking about the possible effects of overlooked variables. But in these time-series examples we are talking about the possible effects of overlooked events. It’s difficult to say which type of problem is more intractable, but they do seem to be two different types of problem.

Analyzing Causal Patterns

When scientists use time series to study the effects of one variable on another, they usually have at least two time series–one for the independent variable and one for the dependent–as in our earlier example on the relation between unemployment and crime. The problems in analyzing causal patterns are difficult but not impossible.

One problem with such research is that because the observations within each series are not independent of each other, the probability of finding a high correlation between the two series may be higher than is suggested by standard formulas. Later we describe a solution to this problem.

A second problem is that it is rarely reasonable to assume that the time sequence of the causal patterns matches the time periods in the study. Thus if increased unemployment typically produced an increase in crime exactly six months later but not five months later, then it would be fairly easy to discover that relationship by correlating monthly changes in unemployment with monthly changes in crime six months later. However, it is much more plausible to assume that increased unemployment in January produces a slight rise in crime during February, a further slight rise during March, and so on for several months. Such effects can be much more difficult to detect, though later we do suggest a solution to this problem.

A third problem in analyzing causal patterns is the familiar problem that correlation does not imply causation. As in ordinary regression problems, it helps to be able to control statistically for covariates. Later we describe one way to do this in time-series problems.

When quantitative data are arranged in the order of their occurrence, the resulting statistical series is called a time series. The quantitative values are usually recorded over equal time interval daily, weekly, monthly, quarterly, half yearly, yearly, or any other time measure. Monthly statistics of Industrial Production in India, Annual birth-rate figures for the entire world, yield on ordinary shares, weekly wholesale price of rice, daily records of tea sales or census data are some of the examples of time series. Each has a common characteristic of recording magnitudes that vary with passage of time.

Time series are influenced by a variety of forces. Some are continuously effective other make themselves felt at recurring time intervals, and still others are non-recurring or random in nature. Therefore, the first task is to break down the data and study each of these influences in isolation. This is known as decomposition of the time series. It enables us to understand fully the nature of the forces at work. We can then analyse their combined interactions. Such a study is known as time-series analysis.

Components of time series

A time series consists of the following four components or elements:

1. Basic or Secular or Long-time trend;
2. Seasonal variations;
3. Business cycles or cyclical movement; and
4. Erratic or Irregular fluctuations.

These components provide a basis for the explanation of the past behaviour. They help us to predict the future behaviour. The major tendency of each component or constituent is largely due to casual factors. Therefore a brief description of the components and the causal factors associated with each component should be given before proceeding further.

1. Basic or secular or long-time trend: Basic trend underlines the tendency to grow or decline over a period of years. It is the movement that the series would have taken, had there been no seasonal, cyclical or erratic factors. It is the effect of such factors which are more or less constant for a long time or which change very gradually and slowly. Such factors are gradual growth in population, tastes and habits or the effect on industrial output due to improved methods. Increase in production of automobiles and a gradual decrease in production of foodgrains are examples of increasing and decreasing secular trend.

All basic trends are not of the same nature. Sometimes the predominating tendency will be a constant amount of growth. This type of trend movement takes the form of a straight line when the trend values are plotted on a graph paper. Sometimes the trend will be constant percentage increase or decrease. This type takes the form of a straight line when the trend values are plotted on a semi-logarithmic chart. Other types of trend encountered are “logistic”, “S-curyes”, etc.
Properly recognising and accurately measuring basic trends is one of the most important problems in time series analysis. Trend values are used as the base from which other three movements are measured.
Therefore, any inaccuracy in its measurement may vitiate the entire work. Fortunately, the causal elements controlling trend growth are relatively stable. Trends do not commonly change their nature quickly and without warning. It is therefore reasonable to assume that a representative trend, which has characterized the data for a past period, is prevailing at present, and that it may be projected into the future for a year or so.

2. Seasonal Variations: The two principal factors liable for seasonal changes are the climate or weather and customs. Since, the growth of all vegetation depends upon temperature and moisture, agricultural activity is confined largely to warm weather in the temperate zones and to the rainy or post-rainy season in the torried zone (tropical countries or sub-tropical countries like India). Winter and dry season make farming a highly seasonal business. This high irregularity of month to month agricultural production determines largely all harvesting, marketing, canning, preserving, storing, financing, and pricing of farm products. Manufacturers, bankers and merchants who deal with farmers find their business taking on the same seasonal pattern which characterise the agriculture of their area.
   The second cause of seasonal variation is custom, education or tradition. Such traditional days as Dewali, Christmas. Id etc., product marked variations in business activity, travel, sales, gifts, finance, accident, and vacationing.

The successful operation of any business requires that its seasonal variations be known, measured and exploited fully. Frequently, the purchase of seasonal item is made from six months to a year in advance. Departments with opposite seasonal changes are frequently combined in the same firm to avoid dull seasons and to keep sales or production up during the entire year. Seasonal variations are measured as a percentage of the trend rather than in absolute quantities. The seasonal index for any month (week, quarter etc.) may be defined as the ratio of the normally expected value (excluding the business cycle and erratic movements) to the corresponding trend value. When cyclical movement and erratic fluctuations are absent in a lime series, such a series is called normal. Normal values thus are consisting of trend and seasonal components. Thus when normal values are divided by the corresponding trend values, we obtain seasonal component of time series.
3. Business Cycle: Because of the persistent tendency for business to prosper, decline, stagnate recover; and prosper again, the third characteristic movement in economic time series is called the business cycle. The business cycle does not recur regularly like seasonal movement, but moves in response to causes which develop intermittently out of complex combinations of economic and other considerations. When the business of a country or a community is above or below normal, the excess deficiency is usually attributed to the business cycle. Its measurement becomes a process of contrast occurrences with a normal estimate arrived at by combining the calculated trend and seasonal movements. The measurement of the variations from normal may be made in terms of actual quantities or it may be made in such terms as percentage deviations, which is generally more satisfactory method as it places the measure of cyclical
tendencies on comparable base throughout the entire period under analysis.
4. Erratic or Irregular Component: These movements are exceedingly difficult to dissociate quantitatively from the business cycle. Their causes are such irregular and unpredictable happenings such as wars, droughts, floods, fires, pestilence, fads and fashions which operate as spurs or deterrents upon the progress of the cycle. Examples such movements are : high activity in middle forties due to erratic effects of 2nd world war, depression of thirties throughout the world, export boom associated with Korean War in 1950.
The common denominator of every random factor it that does not come about as a result of the ordinary operation of the business system and does not recur in any meaningful manner.
Mathematical Statement of the Composition of Time Series
A time series may not be affected by all type of variations. Some of these type of variations may affect a few time series, while the other series may be effected by all of them. Hence, in analysing time series, these effects are isolated. In classical time series analysis it is assumed that any given observation is made up of trend, seasonal, cyclical and irregular movements and these four components have multiplicative relationship.
Symbolically :

O = T × S × C × I
where O refers to original data,
T refers to trend.
S refers to seasonal variations,
C refers to cyclical variations and
I refers lo irregular variations.
This is the most commonly used model in the decomposition of time series.
There is another model called Additive model in which a particular observation in a time series is the sum of these four components.
O = T + S + C + I

Classical Probability: There are ‘n’ number of events and you can find the probability of the happening of an event by applying basic probability formulae. For example – the probability of getting a head in a single toss of a coin is 1/2. This is Classical Probability.

Empirical Probability: This type of probability is based on experiments. Say, we want to know that how many times a head will turn up if we toss a coin 1000 times. According to the Traditional approach, the answer should be 500. But according to Empirical approach, we’ll first conduct an experiment in which we’ll toss a coin 1000 times and then we can draw our answer based on the observations of our experiment.

Addition Theorem on probability:

If A and B are any two events then the probability of happening of at least one of the events is defined as P(AUB) = P(A) + P(B)- P(A∩B).

Since events are nothing but sets,

From set theory, we have

n(AUB) = n(A) + n(B)- n(A∩B).

Dividing the above equation by n(S), (where S is the sample space)

n(AUB)/ n(S) = n(A)/ n(S) + n(B)/ n(S)- n(A∩B)/ n(S)

Then by the definition of probability,

P(AUB) = P(A) + P(B)- P(A∩B).

Example:

If the probability of solving a problem by two students George and James are 1/2 and 1/3 respectively then what is the probability of the problem to be solved.

Solution:

Let A and B be the probabilities of solving the problem by George and James respectively.

Then P(A)=1/2 and P(B)=1/3.

The problem will be solved if it is solved at least by one of them also.

So, we need to find P(AUB).

By addition theorem on probability, we have

P(AUB) = P(A) + P(B)- P(A∩B).

P(AUB) = 1/2 +.1/3 – 1/2 * 1/3 = 1/2 +1/3-1/6 = (3+2-1)/6 = 4/6 = 2/3

Note:

If A and B are any two mutually exclusive events then P(A∩B)=0.

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

Multiplication theorem on probability:

If A and B are any two events  of a sample space such that P(A) ≠0 and P(B)≠0, then

P(A∩B) = P(A) * P(B|A) = P(B) *P(A|B).

Example:  If P(A) =  1/5  P(B|A) =  1/3  then what is P(A∩B)?

Solution: P(A∩B) = P(A) * P(B|A) = 1/5 * 1/3 = 1/15

INDEPENDENT EVENTS:

Two events A and B are said to be independent if there is no change in the happening of an event with the happening of the other event.

i.e. Two events A and B are said to be independent if

P(A|B) = P(A) where P(B)≠0.

P(B|A) = P(B) where P(A)≠0.

i.e. Two events A and B are said to be independent if

P(A∩B) = P(A) * P(B).

Example:

While laying the pack of cards, let A be the event of drawing a diamond and B be the event of drawing an ace.

Then P(A) =  13/52 = 1/4 and P(B) =  4/52=1/13

Now, A∩B = drawing a king card from hearts.

Then P(A∩B) =  1/52

Now, P(A/B) = P(A∩B)/P(B) = (1/52)/(1/13) = 1/4 = P(A).

So, A and B are independent.

[Here, P(A∩B) = =    = P(A) * P(B)]

Note:

(1)    If 3 events A,B and C are independent the

P(A∩B∩C) = P(A)*P(B)*P(C).

(2)    If A and B are any two events, then P(AUB) = 1-P(A’)P(B’).

In our day to day life the “probability” or “chance” is very commonly used term. Sometimes, we use to say “Probably it may rain tomorrow”, “Probably Mr. X may come for taking his class today”, “Probably you are right”. All these terms, possibility and probability convey the same meaning. But in statistics probability has certain special connotation unlike in Layman’s view.

The theory of probability has been developed in 17th century. It has got its origin from games, tossing coins, throwing a dice, drawing a card from a pack. In 1954 Antoine Gornband had taken an initiation and an interest for this area.

After him many authors in statistics had tried to remodel the idea given by the former. The “probability” has become one of the basic tools of statistics. Sometimes statistical analysis becomes paralyzed without the theorem of probability. “Probability of a given event is defined as the expected frequency of occurrence of the event among events of a like sort.” (Garrett)

The probability theory provides a means of getting an idea of the likelihood of occurrence of different events resulting from a random experiment in terms of quantitative measures ranging between zero and one. The probability is zero for an impossible event and one for an event which is certain to occur.

Approaches of Probability Theory

1. Classical Probability:

The classical approach to probability is one of the oldest and simplest school of thought. It has been originated in 18th century which explains probability concerning games of chances such as throwing coin, dice, drawing cards etc.

The definition of probability has been given by a French mathematician named “Laplace”. According to him probability is the ratio of the number of favourable cases among the number of equally likely cases.

Or in other words, the ratio suggested by classical approach is:

Pr. = Number of favourable cases/Number of equally likely cases

For example, if a coin is tossed, and if it is asked what is the probability of the occurrence of the head, then the number of the favourable case = 1, the number of the equally likely cases = 2.

Pr. of head = 1/2

Symbolically it can be expressed as:

P = Pr. (A) = a/n, q = Pr. (B) or (not A) = b/n

1 – a/n = b/n = (or) a + b = 1 and also p + q = 1

p = 1 – q, and q = 1 – p and if a + b = 1 then so also a/n + b/n = 1

In this approach the probability varies from 0 to 1. When probability is zero it denotes that it is impossible to occur.

If probability is 1 then there is certainty for occurrence, i.e. the event is bound to occur.

Example:

From a bag containing 20 black and 25 white balls, a ball is drawn randomly. What is the probability that it is black.

Pr. of a black ball = 20/45 = 4/9 = p, 25 Pr. of a white ball = 25/45 = 5/9 = q

p = 4/9 and q = 5/9 (p + q= 4/9 + 5/9= 1)

2. Relative Frequency Theory of Probability:

This approach to probability is a protest against the classical approach. It indicates the fact that if n is increased upto the ∞, we can find out the probability of p or q.

Example:

If n is ∞, then Pr. of A= a/n = .5, Pr. of B = b/n = 5

If an event occurs a times out of n its relative frequency is a/n. When n becomes ∞, is called the limit of relative frequency.

Pr. (A) = limit a/n

where n → ∞

Pr. (B) = limit bl.t. here → ∞.

Axiomatic approach

An axiomatic approach is taken to define probability as a set function where the elements of the domain are the sets and the elements of range are real numbers. If event A is an element in the domain of this function, P(A) is the customary notation used to designate the corresponding element in the range.

Probability Function

A probability function p(A) is a function mapping the event space A of a random experiment into the interval [0,1] according to the following axioms;

Axiom 1. For any event A, 0 ≤ P(A) ≤ 1

Axiom 2. P(Ω) = 1

Axiom 3. If A and B are any two mutually exclusive events then,

                              P(A ∪ B)) = P(A) + P(B)

As given in the third axiom the addition property of the probability can be extended to any number of events as long as the events are mutually exclusive. If the events are not mutually exclusive then;

P(A ∪ B) = P(A) + P(B) – P(A∩B)

P(A∩B) is Φ if both the events are mutually exclusive.

If there are two types of objects among the objects of similar or other natures then the probability of one object i.e. Pr. of A = .5, then Pr. of B = .5