DEFINITIONS

 Experiment

 An experiment is a set of processes terms, which are carried out under stipulated conditions to study the phenomena associated with it. Broadly, there can be two types of experiments:

(i)     Experiments with definite outcome: These type of experiments are certain in nature. Outcomes of experiments are known in advance.

(ii)    Experiments with indefinite outcome: These type of experiment are uncertain in nature, we cannot predict the outcome with certainty.

 Random Experiment

An experiment whose all possible outcomes (results) are known in advance, but the result of any specific performance cannot be predicted before completion of the experiment.

Illustration:

        Consider the following experiments.

(i)     Tossing a coin

(ii)    Rolling a die

(iii)    Drawing a card from a pack of well-shuffled pack of 52 cards.

Solution:

        All the experiments have more than one possible outcome. All outcomes are known in advance. Hence all these experiments are random experiments.

Sample Space

The set of all possible outcomes of an experiment (or trial) is called the sample space. It is usually denoted by s.

A die is thrown. The upper face can show any of the six numbers (1, 2, 3, 4, 5, 6). Due to this, we get a sample space consisting of 6 elements.

        A card is drawn from a well-shuffled pack of cards. We get a sample space consisting of 52 elements.

For example, for the experiment of tossing a coin, the sample space is a set S = {head, tail}. For the experiment of tossing two coins, the sample space is S = {head - head, head - tail, tail - head, tail - tail}.

Consider the experiment of shooting a tiger until there is a hit. The sample space is the countably infinite set S = {h, mh, mmh, ...} where h denotes a hit, m denotes a miss and mh denotes a miss followed by a hit and so on.

Event

The appearance of a particular outcome, which we may find in the sample space, is an event. Any subset of the sample space is called an event.

Choosing an ace from a pack of well-shuffled cards is an event. This event occurs 4 times in a pack of card i.e. in sample space. Observing an even number when a dice is thrown is an event. This event occurs three times in the sample space of 6 elements. So, event is a subset of sample space.

Complement of an Event:

        Let S be the sample space and A be some events, then the set of all out comes which are in S but not in A is called the complete of event A. It is denoted by A^- or A^c.

Illustration:

        In rolling a die event A is described as numbers

        A = {2, 4, 6}

Mutually Exclusive Events:

        Let the two events be A and B. If occurrence of event A excludes the possibility of occurrence of B and vice-versa, then we say that A and N are mutually exclusive event i.e. they cannot occur simultaneously. It (i.e. mutual exclusion) can happen only when the sets of two events have no common point.

Illustration:

        In rolling a die, event A is described as appearance of an even number and event B is described as appearance of an odd number.

        A = {2, 4, 6}

        B = {1, 3, 5}

        Clearly if one says that A has occurred we will instantly conclude that B cannot occur (in that trial of course). So,

        A ∩ B = Φ

Equally likely Events:

        Two or more events are said to be equally likely when there is no reason to prefer one event over other i.e. they have equal (not same) number of points in their sets.

Illustration:

        In rolling a die event A is described as number showing less than 4 and event B is described as number appearing greater than 3.

        A = {1, 2, 3}

        B = {4, 5, 6}

        Clearly A and B have equal number of points in the sample space and hence are equally likely.

Exhaustive Events:

        If n events A₁, A₂, ........., A_n related to any particular sample space are such that if we take union of the sets of all the n events, sample space is formed. i.e.

        A₁ υ A₂ υ A₃ υ ............... υ A_n = S

Illustration:

        In rolling a die three events are described as follows

                A₁ : even number appears

                A₂ : less than 4 appears

                A₃ : greater than 4 appears

        Hence A₁ : {2, 4, 6}

                 A₂ : {1, 2, 3}

                 A₃ : {5, 6}

A₁ υ A υ A₃ = {1, 2, 3, 4, 5, 6} which is the sample space for the experiment of rolling a die.

Mutually Exclusive and Exhaustive Events.

        Events A₁, A₂, .......... A_n are said to be mutually exclusive and exhaustive if they satisfy the condition for mutual exclusion and exhaustiveness both.

        i.e.    A₁ υ A₂ υ A₃ ............ υ A_n = S

        and   A_i ∩ A_j = Φ, where i = 1, 2, ........., n

                                        j = 1, 2, ........... n and i ≠ j.

        i.e. mutually exclusive and exhaustive events are those events of which union is equal to sample space and occurrence of any one of them excludes the possibility of occurrence of all others.

Illustration:

        In rolling a die, events A₁, A₂, A₃ and A₄ are described as follows

                A₁ : Number less then 2 occurs.

                A₂ : 2 occurs

                A₃ : odd number greater that 1 occur.

                A₄ : even number greater than 2 occur.

                A₁ : {1}

                A₂ : {2}

                A₃ : {3, 5}

                A₄ : {4, 6}

        Clearly, A₁ υ A₂ υ A₃ υ A₄ = {1, 2, 3, 4, 5, 6} = S

and   A₁ ∩ A₂ = A₁ ∩ A₃ = A₁ ∩ A₄ = A₂ ∩ A₃ = A₂ ∩ A₄ = A₃ ∩ A₄ = Φ

        i.e. A_i ∩ A_j = Φ, i ≠ j, i, j = 1, 2, 3, ............ n.

Independent Events

        If the occurrence or non occurrence of any event does not affect the sample space of occurrence of the other, then two events are said to independent.

Equally likely events/outcomes

In an experiment, two or more event/outcomes are said to be equally likely, if they have the same chances associated with them. i.e. no one of them has more chance of occurrence than others.