Module 2. Theory of probability

Lesson 6

ELEMENTARY NOTIONS OF PROBABILITY

6.1  Introduction

If an experiment is repeated under essentially homogenous and identical conditions then we may come across two types of situations viz., (i) the result or outcome is unique or certain (ii) the result or outcome may be one of the several possible outcomes. The phenomena covered by the former situation (i) is known as ‘deterministic’ or ‘predictable’ while the phenomena covered by situation (ii) is known as ‘Unpredictable’ or ‘Probabilistic’. By a deterministic phenomenon we mean that the result can be predicted with certainty e.g. Boyle’s law stating that PV = RT = Constant provided temperature remains the same; Newton’s first law of motion stating that v = u + at, where u is the initial velocity and a is the acceleration; Ohm’s law stating that C = E / R, where C is the flow of current, E is the potential difference between two ends of the conductor and R is the resistance. On the other hand by the probabilistic phenomena we mean that result is not sure e.g. in tossing a coin we are not sure as to whether head or tail will be obtained. Similarly, if any electronic equipment has worked for certain number of hours, nothing can be said about its further performance as to it may fail to function any moment. In such situations, the word ‘probability’ has relevance. In day to day life, we all make use of the word ‘probability’, but generally people have no definite idea about the meaning of probability. For example, we often hear or talk phrases like “the fat content of milk sample obtained from buffalo is likely to be 5.5 percent”, “the daily milk yield of this cow is likely to be more than 15 kg”, “there is a chance that it may rain today” or “India may win this match” or “milk production in India is likely to be 200 million tonnes by 2030”. In all the above statements, the terms probably, likely, chance etc. convey the same meaning i.e. the events are not certain to take place. In other words, there is involved an element of uncertainty or chance in all these cases. A numerical measure of uncertainty is provided by the theory of probability.’ The theory of probability came into existence when problems of games were referred to mathematicians like B. Pascal, P. Fermat, James Bernoulli, De-Moivre, Karl Pearson, Laplace and others. Later, the classical theory of probability was given by R.A. Fisher. Von-Mises introduced the empirical approach to the theory of probability through the notion of sample space. The idea of axiomatic approach was originated by A. Kolmogorov. But in modern times, it has acquired a great importance in decision making.

6.2  Basic Concept

Before the definition of the word probability is given, it is necessary to define the following basic concepts and terms widely used in its study:

6.2.1  Random experiment

An experiment is said to be a random experiment if when conducted repeatedly under essentially homogeneous conditions, the result is not unique but may be anyone of the various possible outcomes. In other words an experiment whose outcomes can’t be predicted in advance is called a random experiment.  For instance, if a fair coin is tossed three times, it is possible to enumerate all the possible eight sequences of head (H) and tail (T). But it is not possible to predict which sequence will occur at any time.

6.2.2  Sample space

The set of all possible outcomes of a random experiment is known as the sample space and is denoted by S. Each conceivable outcome of a random experiment under consideration is called a sample point. The totality of all conceivable sample points is called a sample space for example sample space of a trial conducted by tossing of three coins is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. In the above experiment it is simple to note that anyone sequence of H and/or T is a sample point whereas all the possible eight sample points constitute the sample space.

6.2.3  Trial and event

Performing a random experiment is called a trial and outcome or combinations of outcomes are termed as ‘Events’ or ‘Cases’. Any subset of the sample space is an event. In other words, the set of sample points which satisfy certain requirement(s) is called an event. For example, if a coin is tossed repeatedly, the result is not unique. Tossing of coin is a random experiment and getting a head or tail is an event.

6.2.4  Exhaustive events

It is defined as total number of all possible outcomes of any trial. In other words, if all the possible outcomes of an experiment are taken into consideration, then such events are called exhaustive events e.g. when a coin is tossed three times there are eight exhaustive events and when two dice are thrown then exhaustive events are 36.

6.2.5  Favourable events

The numbers of outcomes of a random experiment which result in the happening of an event are termed as the cases favourable to the event. For example in three tossing of a coin, the cases favourable to the event that there are exactly two heads is 3, viz. HTH, HHT and THH and for getting at least two heads is 4 , viz. HTH,THH, HHT and HHH.

6.2.6  Mutually exclusive events

Two or more events are said to be mutually exclusive if the happening of one of them prevents or precludes the happening of the all others in the same experiment. Two events E₁ and E₂ are said to be mutually exclusive when they can’t happen simultaneously in a single trial. In other words if there is no sample point in E₁ which is common to the sample point in E₂ i.e. E₁∩E₂=f, the events E₁ and E₂ are said to be mutually exclusive. In tossing a die, the events 1, 2, 3, 4, 5 and 6 are mutually exclusive events because all the six events cannot happen simultaneously in a single trial. If it shows 3, then the event of getting 3 precludes the event of getting 1, 2, 4, 5, 6 at the same time.  

6.2.7  Complementary event

The complement of an event A, means non-occurrence of an event A and is denoted by Ā or A^c. A^c/Ā contains those points of the sample space which do not belong to A. In tossing a coin, occurrence of Head (H) and Tail (T) are complementary events. In tossing of a die, occurrence of an even number (2, 4, 6) and odd number (1, 3, 5) are complementary events.

6.2.8  Independent events

Events are said to be independent of each other if happening of any one of them is not affected by and does not affect the happening of any one of others. In other words two or more events are said to be independent if the happening (or non happening) of any one does not depend on the happening or non-happening of any other, otherwise they are said to be dependent. For example

a) In tossing an unbiased coin, the event of getting a head in the first toss is independent of getting a head in the second, third and subsequent throws.

b) If we draw a card from a pack of well-shuffled cards and replace it before the second card is drawn, the result of second draw is independent of the first draw. But if the first card drawn is not replaced then the second draw is dependent on the first draw.

c) A bag contains balls of two different colours say red and white. The two balls are drawn successively. First a ball is drawn from one bag and replaced after noting its colour. Let us suppose that it is white and is denoted by event E₁. Another ball is drawn from the same bag E₂. The result of the second draw is independent of the first draw. Hence the events E₁ and E₂ are independent.

6.2.9  Equally likely cases

The events are said to be equally likely if the chance of happening of each event is equal or same. In other words, cases are said to be equally likely when one does not occur more often than the others e.g. if a die is rolled, any face is as likely to come up as any other face. Hence, the six outcomes 1, 2, 3, 4, 5 or 6 appearing up are equally likely.

6.2.10  Simple (Elementary) events

An event which contains only a single sample point is called an elementary event or simple event e.g. in tossing a die, getting a number 5 is called a simple event.

6.2.11  Compound events

When two or more events occur in connection with each other, their simultaneous occurrence is called a compound event. Joint occurrence of two or more events is called a compound event. An event is termed compound if it represents two or more simple events e.g. if a bag contains 4 white and 3 black balls. If we are required to find a chance in which 3 balls drawn are all white is a simple event. However if we are required to find out the chance of drawing 3 white and then 2 black balls, we are dealing with a compound event because it is made up of two events.

6.3  Definition of Probability

The chance of happening of an event when expressed quantitatively is called probability. The probability is defined in the following three different ways:

· Classical, Mathematical or ‘a Priori’ definition.

· Empirical, Relative or Statistical definition

· Axiomatic definition  

6.3.1  Classical or Mathematical definition of probability

This is the oldest and simplest definition of probability. This definition is based on the assumption that the outcomes or results of an experiment are equally likely and mutually exclusive. According to James Bernoulli who was the first man to obtain a quantitative measure of uncertainty “If a random experiment results in N exhaustive, mutually exclusive and equally likely cases out of which m are favourable to the happening of an event A”, then probability of occurrence of A, usually denoted by  P(A) is given by

Example 1. Two identical symmetric dice are thrown. Find the probability of obtaining a total score of 8.

The total number of possible outcomes is 6x6=36. There are 5 sample points (2, 6), (3, 5) (4, 4), (5, 3), (6, 2), which are favourable to the event A of getting a total score of 8. Hence, the required probability is 5/36.

Properties:

· The number of cases favourable to the complimentary event i.e. non-happening of event A are (N-m)and by definition of probability of non–occurrence of A is given by:

· Since m and N are non-negative integers, P(A)≥0. Further, since the favourable number of cases to A are always less than total number of cases N, i.e. m ≤ N, we have P (A) ≤1. Hence the probability of any event is a number lying between 0 and 1 i.e.  0≤ P (A) ≤1. If P(A)=0 then this event is said to be impossible event. If P (A) =1, then A is called a certain event.

The above definition of probability is widely used, but it cannot be applied under the following situations:

· If it is not possible to enumerate all the possible outcomes for an experiment.

· If the sample points (outcomes) are not mutually independent.

· If the total number of outcomes is infinite.

· If each and every outcome is not equally likely.

It is clear that the above drawbacks of a classical approach restrict its use in practical problems. Yet this is still widely used for problems concerning the tossing of coin(s), throwing of dice, game of cards and selection of balls of different colours from the bag etc.

The probability by classical approach cannot be discovered in the cases where situations like an electric bulb will fuse before it is used for 100 hours, a patient will die if operated for an ailment, a student will fail in a particular examination, a rail compartment in which you are travelling will catch fire, a fan will fall on you while sitting under fan etc. under such circumstances another definition can be used.

6.3.2  Statistical definition of probability

If an experiment is performed repeatedly under essentially homogeneous and identical conditions, then the limiting value of the ratio of the number of times the event occurs to the number of trials, as the number of trials becomes indefinitely large, is called the probability of happening of the event, assuming that the limit is finite and unique. Let an event A occurs m times in N repetitions of a random experiment. Then the ratio m/N gives the relative frequency of the event A. When N becomes sufficiently large, it is called the probability of A.

The above definition of probability involves a concept which has long term consequences. This approach was initiated by Von Mises. Moreover N is not equal to infinity. Thus, in this case, the probability is the limit of relative frequency. Whether such a limit always exists, is not definite. Hence statistical definition of probability is also not very sound.

The two definitions of probability are apparently different. In the ‘a prior’ definition, it is the relative frequency of favourable cases to the total number of cases. Since in the relative frequency approach, the probability is obtained objectively by repetitive empirical observations, hence it is known as empirical probability. The empirical definition provides validity to the classical theory of probability.

6.3.3  Axiomatic approach to probability

The modern theory of probability is based on the axiomatic approach introduced by the Russian Mathematician A.N. Kolmogorov in 1930’s. The axiomatic definition of probability includes both the classical and empirical definition of probability and at the same time is free from their drawbacks. It is based on certain properties or postulates, commonly known as axiom, which are defined from these axioms alone the entire theory is developed by logic of deduction. It is defined as given a sample space of a random experiment, the probability of the occurrence of any event A is defined as asset function P(A)satisfying the following axioms

a) P (A) is defined, is real and non-negative i.e. P (A) > 0.

b) The probability of entire sample space is one i.e. P(S) = 1.

c) If A₁, A₂, ---, A_n are mutually exclusive events, then the probability of the occurrence of either  A₁or A₂, ---or A_n denoted by P(A₁ ∪ A₂ ∪ --- ∪ A_n) = P(A₁)+P(A₂)+---+P( A_n) .

The above axioms are known as axioms of positiveness, certainty and unity respectively.

Probability in this approach is defined as, let S be the sample space of a random experiment with large number of sample points N i.e. n(S) =N. Let the number of occurrences (sample points) favourable to the event A be denoted by n(A). Then the probability of an event A is equal to

6.4  Calculation of probability of an event

 The probability of an event can be calculated by the following methods:

Method I: Find the total number of exhaustive cases (N). Thereafter, obtain the number of favourable cases to the event i.e. m. Divide the number of favourable cases by the total number of equally likely cases. This will give the probability of an event. The following example will illustrate this.

Example 2.  Two dice are tossed. Find the probability that the sum of dots on the faces that turn up is a) 8 and b) 11.

Solution:  When two dice are tossed total number of possible outcomes =36

a) Number of outcomes to get a sum of 8 are (6, 2) (5, 3), (4, 4), (3, 5) and (2, 6) i.e. the number cases favourable to this event is equal to 5. Hence, probability of getting a sum of 8 when two dice are thrown =

b) Number of outcomes to get a sum of 11 are (6, 5) and (5, 6) i.e. the number cases favourable to this event is equal to 2. Hence, probability of getting a sum of 11 when two dice are thrown =

Example 3: From a herd containing 5 Karan Fries and 4 Sahiwal cows, a cow is selected at random. What is the probability that it is a Sahiwal Cow?

Solution: Total number of cows in the herd = 5+4 = 9

                Number of Sahiwal cows = 4

                Probability of getting a Sahiwal cow =

Method II: The Fundamental Principle or the Fundamental Rule of Counting:

If one operation can be performed in m different ways and another operation can be performed in n different ways, then the two operations when associated together can be performed in m x n ways.

Method III: Use of Permutation and Combination in Theory of Probability:

Permutation:

The word permutation in simple language means arrangement. A permutation denoted by P is an arrangement of a set of objects in a definite order:

Combination:

The concept of combination is very useful in understanding theory of probability. It is not always possible that the number of cases favourable to the happening of an event is easily determined. In such cases the concept of combination is used. The different selections that can be made out of a given set of things taking some or all of them at a time are called combinations. The combinations of n things, taking r at a time is denoted by ⁿC_r , symbolically

Example 4: From a pack of 52 cards, two cards are drawn at random. Find the probability that one is a king and the other is queen.

Solution: Two cards can be drawn from 52 cards in ⁵²C₂ ways.

There are 4 kings and 4 queens in a pack of cards. A king can be drawn in ⁴C₁ ways and a queen can be drawn in ⁴C₁ ways.

The probability of getting a king card and the other a queen card is

Example 5: A herd consists of 5 Karan Fries and 6 Sahiwal cows. Two cows are chosen at random from this herd. What is the probability that

a) One is Karan Fries and other is Sahiwal.

b) Both are Sahiwal.

Solution:  Total number of cows in the herd =5+6=11

Two cows can be chosen from 11 cows in

a) In the herd there are 5 Karan Fries and 6 Sahiwal cows. One Karan Fries out of 5 can be chosen in

One Sahiwal cow out of 6 can be chosen in 

The probability of choosing one Karan Fries and one Sahiwal cow is

b) Both the cows are Sahiwal can be chosen in 

The probability of choosing both the Sahiwal cows is