# Probability - A Theoretical Approach

The word â€˜probabilityâ€™ is commonly used in our everyday conversation and we generally use this word even without going into the details of its actual meaning. Generally, people have a rough idea about its meaning. In our day-to-day life we come across statements like :(i) Probably it may rain today.

(ii) Indian Cricket team has good chance of winning the next Word Cup.

(iii) He is probably correct.

In such statements, we generally use the terms: possible, probable, chance, likely, etc. All these terms convey the same sense that the event is not certain to take place or, in other words, there is uncertainty about the occurrence (or happening) of the event in question. Thus, the word â€˜probabilityâ€™, relates to uncertainty about what has happened or what is going to happen. However, in the theory of probability we assign numerical value to the degree of uncertainty.

The concept of probability originated in the beginning of eighteenth century in problems pertaining to games of chance such as throwing a die, tossing a coin, drawing a card from a pack of cards, etc. Starting with games of chance, â€˜probabilityâ€™ today has become one of the basic tools of Statistics and has wide range of applications in Science and Engineering.

In this chapter, we shall introduce the concept of probability as a measure of uncertainty.

In the theory of probability we deal with events, which are outcomes of an experiment. The word â€˜experimentâ€™ means an operation which can produce some well defined outcome(s).

There are two types of experiments:

(i) Deterministic

(ii) Random or Probabilistic.

# Deterministic experiments

**Deterministic experiments**

**are those experiments which when repeated under identical conditions produce the same result or outcome. When experiments in science and engineering are repeated under identical conditions, we obtain almost the same result every time.**

**Example : **The experiments that we conduct to verify the laws of science are the best example for these.

An Experiment conducted to verify the Newton's Laws of Motion.

If an experiment, when repeated under identical conditions, do not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, then it is known as a random or probabilistic experiment.

**Example : **A pointer has 4 equal sectors colored yellow, blue, green and red. The chances of landing on red after spinning are a probabilistic experiment.

**Example : **in tossing of a coin one is not sure if a head or a tail will be obtained, so it is a random experiment. Similarly, rolling an unbiased die and drawing a card from a well-shuffled pack of cards are examples of a random experiment.

Throwing of a die is also a random experiment as any of the six faces of the die may come up. In this experiment, there are six possibilities (1 or 2 or 3 or 4 or 5 or 6).

# Elementary Event

An outcome of a random experiment is called an Elementary Event.Consider the random experiment of tossing of a coin.

The possible outcomes of this experiment are head (H) or tail (T).

Thus, if we define E_{1} = Getting head (H) on the upper face of the coin,

and,

E_{2} = Getting tail (T) on the upper face of the coin

Then, E_{1} and E_{2} are elementary events associated with the experiments of tossing of a coin.

Let us consider the random experiment of tossing two coins simultaneously. The possible outcomes of this experiment are as under

If we define

HH = Getting heads on both the coins,

HT = Getting head on the first and tail on the second,

TH = Getting tail on the first and head on the second,

and, TT = Getting tails on both coins.

Then, HH, HT, TH and TT are elementary events associated with the random experiment of tossing of two coins.

Similarly, if three coins are tossed simultaneously, then the elementary events associated with this experiment are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.

Consider a cubical die marked with numbers, 1, 2, 3, 4, 5 and 6 on its six faces. Consider now the random experiment of throwing a cubical die. If the die is rolled, then any one of the six faces may come upward. So, there are six possible outcomes of this experiment, namely, 1, 2, 3, 4, 5, 6.

Thus, if we define E_{i} = Getting a face marked with number i, where i = 1, 2, â€¦ , 6.

Then, E_{1}, E_{2 }, ... E_{6 }are six elementary events associated to this experiment.

In this experiment, elementary event E_{i} is generally denoted by i, where i = 1, 2, 3 â€¦6.

Now, consider the random experiment in which two six faced dice are rolled together or a die is rolled twice. If (i, j) denotes the outcome of getting number i on first die and number j on second die, then possible outcomes of this experiment are

{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}.

Clearly, these outcomes are elementary events associated with the random experiment of throwing two six faced dice together. The total number of these elementary events is 36.

If a card is drawn from a well-shuffled pack of 52 cards, then any one of 52 cards can be the outcome. So there are 52 elementary events associated to the random experiment of drawing a card from a pack of 52 playing cards.

**Sure event**

An event that has a 100% chance of occurrence whenever the experiment is performed is a sure event.

**Example : **Throwing two dice simultaneously, the event of getting a sum less than 13 is a sure event.

A sure event is also called a certain event.

An event that has a 0% chance of occurrence whenever the experiment is performed is an impossible event.

**Example : **Throwing two dice simultaneously, the event of getting a sum greater than 13 is an impossible event.

**Simple Event**

An event having only one sample point of a sample space is called a simple event.

**Example**

There are 2 children in a family. Find the events that:

(i) both children are boys

**Solution**

Sample Space = {BB, BG, GB, GG}

(i) Let A be the event that both children are boys. âˆ´A = {BB}

**A is a Simple event**

# Compound Event

An event associated to a random experiment is a compound event if it is obtained by combining two or more elementary events associated to the random experiment.

In a single throw of a die, the event "Getting an even number" is a compound event as it is obtained by combining three elementary events, viz., 2, 4, 6.

Similarly, "Getting an odd number" is a compound event in a single throw of a die.

Consider the random experiment of tossing two coins simultaneously. If we define the event "Getting exactly one head", then HT and TH are two elementary events associated to it. So, it is a compound event.

**Occurrence of an Event **

An event A associated to a random experiment is said to occur if any one of the elementary events associated to the event A is an outcome.

Consider the random experiment of throwing an unbiased die. Let A denote the event "Getting an even number". Elementary events associated to this event are: 2, 4, 6. Now, suppose that in a trial the outcome is 4, then we say that the event A has occurred. In another trial, let the outcome be 3, then we say that the event A has not occurred.

Let a die be rolled and the outcome of the trial be 4. Then, we can say that each of the following events has occurred:

(i) Getting a number greater than or equal to 2;

(ii) Getting a number less than or equal to 5;

(iii) Getting an even number.

On the basis of the same outcome, we can also say that the following events have not occurred:

(i) Getting an odd number;

(ii) Getting a multiple of 3.

Let us now consider the random experiment of throwing a pair of dice. If (2, 6) is an outcome of a trial, then we can say that each of the following events have occurred.

(i) Getting an even number on first die;

(ii) Getting an even number on both dice;

(iii) Getting 8 as the sum of the numbers on two dice.

However, on the basis of the same outcome, one can also say that the following events have not occurred:

(i) Getting a multiple of 3 on first die;

(ii) Getting an odd number on first die;

(iii) Getting a doublet.

**Favourable Elementary Events**

An elementary event is said to be favourable to a compound event A, if it satisfies the definition of the compound event A.

In other words, an elementary event E is favourable to a compound event A, if we say that the event A occurs when E is an outcome of a trial.

Consider the random experiment of throwing a pair of dice and the compound event A defined by "Getting 8 as the sum". We observe that the event A occurs if we get any one of the following elementary events as outcome:

{(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)}

So, there are 5 elementary events favourable to event A.

If two coins are tossed simultaneously and A is an event associated to it defined as "Getting exactly one head", we say that the event A occurs if we get either HT or TH as an outcome. So, there are two elementary events favourable to the event A.

**Negation of an Event**

Corresponding to every event A associated with a random experiment we define an event "not A" which occurs when and only when A does not occur. The event "not A" is called the negation of event A and is denoted by A.

Clearly, event A occurs if and only if A does not occur.

**Probability of an Event**

Probability of an event = .