Introduction of classical theory of probability

In general, to obtain the probability of an event, we find the ratio of the number of outcomes favourable to the event, to the total number of equally likely outcomes. This theory of probability is known as classical theory of probability.

Statistical approach of probability

The probability on the basis of observations and collected data. This is called statistical approach of probability.

Random Experiments

An experiment is called random experiment if it satisfies the following two conditions:
(i) It has more than one possible outcome.
(ii) It is not possible to predict the outcome in advance.

Outcomes and sample space

A possible result of a random experiment is called its outcome.

Sample space of the experiment

Consider the experiment of rolling a die. The outcomes of this experiment are 1, 2, 3, 4, 5, or 6, if we are interested in the number of dots on the upper face of the die. The set of outcomes {1, 2, 3, 4, 5, 6} is called the sample space of the experiment.

Sample space

The set of all possible outcomes of a random experiment is called the sample space associated with the experiment. Sample space is denoted by the symbol S.

Sample point

Each element of the sample space is called a sample point. In other words, each outcome of the random experiment is also called sample point.

Event

Any subset E of a sample space S is called an event.

Occurrence of an event

Consider the experiment of throwing a die. Let E denotes the event “ a number less than 4 appears”. If actually ‘1’ had appeared on the die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3, we say that event E has occurred.
Thus, the event E of a sample space S is said to have occurred if the outcome ωof the experiment is such that ω∈ E. If the outcome ω is such that ω ∉ E, we say that the event E has not occurred.

Types of events

Events can be classified into various types on the basis of the elements they have.
(i) Impossible and Sure Events
(ii) Simple Event
(iii) Compound Event

(i) Impossible and Sure Events

The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called the sure event.

(ii) Simple Event

If an event E has only one sample point of a sample space, it is called a simple (or elementary) event. Example:-
In the experiment of tossing two coins, a sample space is S={HH, HT, TH, TT} There are four simple events corresponding to this sample space. These are E1= {HH}, E2={HT}, E3= { TH} and E4={TT}.

(iii) Compound Event

If an event has more than one sample point, it is called a Compound event. Example:-
In the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc. are all compound events.
The subsets of S associated with these events are
E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}

Complementary Event

For every event A, there corresponds another event A′called the complementary event to A. It is also called the event ‘not A’. For example, take the experiment ‘of tossing three coins’. An associated sample space is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Let A={HTH, HHT, THH} be the event ‘only one tail appears’
Thus the complementary event ‘not A’ to the event A is A′ = {HHH, HTT, THT, TTH, TTT}

The Event ‘A or B’

When the sets A and B are two events associated with a sample space, then ‘A ∪ B’ is the event ‘either A or B or both’. This event ‘A ∪ B’ is also called ‘A or B’.
Therefore Event ‘A or B’ = A ∪ B ={ω : ω ∈ A or ω∈ B}

The Event ‘A and B’

If A and B are two events, then the set A ∩ B denotes the event ‘A and B’.
Thus, A ∩ B = {ω : ω ∈ A and ω ∈ B}
For example, in the experiment of ‘throwing a die twice’ Let A be the event ‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6)} so A ∩ B = {(6,5), (6,6)}

The Event ‘A but not B’

We know that A–B is the set of all those elements which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not B’. We know that A – B = A ∩ B´ .

Mutually exclusive events

In general, two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously.
In this case the sets A and B are disjoint. Again in the experiment of rolling a die, consider the events A ‘an odd number appears’ and event B ‘a number less than 4 appears’
Obviously A = {1, 3, 5} and B = {1, 2, 3} Now 3 ∈ A as well as 3 ∈ B Therefore, A and B are not mutually exclusive events.