Probability
Experiment
An operation which produce some well-defined results or outcomes is
called an experiment.
Types of Experiments
1. Deterministic Experiment
Those experiments, which when repeated under identical conditions
produce the same result or outcome are known as deterministic
experiment.
2. Probabilistic/Random Experiment
Those experiments, which when repeated under identical conditions,
do not produce the same outcome every time but the outcome produced
is one of the several possible outcomes, are called random experiment.
Some Basic Definitions
(i) Trial Performing an experiment is called a trial. The number
of times an experiment is repeated is called the number of trials.
(ii) Sample Space The set of all possible outcomes of a random
experiment is called the sample space of the experiment and it is
denoted by S.
(iii) Sample Point The outcome of an experiment is called the
sample point, i.e. the elements of set S are called the sample
points.
(iv) Event A subset of the sample space associated with a random
experiment is called event or case.
(v) Elementary (or Simple) Event An event containing only one
sample point is called elementary event (or indecomposable
event).
(vi) Compound Event An event containing more than one sample
points is called compound event (or decomposable event).
(vii) Occurrence of an Event An event associated to a random
experiment is said to occur, if any one of the elementary events
associated to it is an outcome.
, (viii) Certain Event An event which must occur, whatever be the
outcomes, is called a certain event (or sure event).
(ix) Impossible Event An event which cannot occur in a random
experiment, is called an impossible event.
(x) Favourable Outcomes Let S be the sample space associated
with a random experiment and E ⊂ S. Then, the elementary
events belonging to E are known as the favourable outcomes to E.
(xi) Equally likely Outcomes The outcomes of a random
experiment are said to be equally likely, when each outcome is as
likely to occur as the other.
Algebra of Events
Let A and B are two events associated with a random experiment,
whose sample space is S. Then,
(i) the event ‘not A’ is the set A′ or S − A
(ii) the events A or B is the set A ∪ B
(iii) the events A and B is the set A ∩ B
(iv) the events A but not B is the set A − B or A ∩ B′
Note For more details, see operations on sets.
Probability—
Theoretical (Classical) Approach
If there are n equally likely outcomes associated with a random
experiment and m of them are favourable to an event A, then the
probability of happening or occurrence of A, denoted by P ( A), is given by
m Number of favourable outcomes to A
P ( A) = =
n Total number of possible outcomes
Axiomatic Approach
Let S = { w1 , w2 , w3 , ... wn } be a sample space, then according to
axiomatic approach we have the following
(i) 0 ≤ P ( wi ) ≤ 1 for each wi ∈ S
(ii) P ( w1 ) + P ( w2 ) + ... + P ( wn ) = 1
(iii) For any event A, P ( A) = ΣP ( wi ), wi ∈ A.
Note
l Theoretical approach is valid only when the outcomes are equally likely and
number of total outcomes is known.
l P(sure event) = P(S ) = 1 and P(impossible event) = P( φ ) = 0
Experiment
An operation which produce some well-defined results or outcomes is
called an experiment.
Types of Experiments
1. Deterministic Experiment
Those experiments, which when repeated under identical conditions
produce the same result or outcome are known as deterministic
experiment.
2. Probabilistic/Random Experiment
Those experiments, which when repeated under identical conditions,
do not produce the same outcome every time but the outcome produced
is one of the several possible outcomes, are called random experiment.
Some Basic Definitions
(i) Trial Performing an experiment is called a trial. The number
of times an experiment is repeated is called the number of trials.
(ii) Sample Space The set of all possible outcomes of a random
experiment is called the sample space of the experiment and it is
denoted by S.
(iii) Sample Point The outcome of an experiment is called the
sample point, i.e. the elements of set S are called the sample
points.
(iv) Event A subset of the sample space associated with a random
experiment is called event or case.
(v) Elementary (or Simple) Event An event containing only one
sample point is called elementary event (or indecomposable
event).
(vi) Compound Event An event containing more than one sample
points is called compound event (or decomposable event).
(vii) Occurrence of an Event An event associated to a random
experiment is said to occur, if any one of the elementary events
associated to it is an outcome.
, (viii) Certain Event An event which must occur, whatever be the
outcomes, is called a certain event (or sure event).
(ix) Impossible Event An event which cannot occur in a random
experiment, is called an impossible event.
(x) Favourable Outcomes Let S be the sample space associated
with a random experiment and E ⊂ S. Then, the elementary
events belonging to E are known as the favourable outcomes to E.
(xi) Equally likely Outcomes The outcomes of a random
experiment are said to be equally likely, when each outcome is as
likely to occur as the other.
Algebra of Events
Let A and B are two events associated with a random experiment,
whose sample space is S. Then,
(i) the event ‘not A’ is the set A′ or S − A
(ii) the events A or B is the set A ∪ B
(iii) the events A and B is the set A ∩ B
(iv) the events A but not B is the set A − B or A ∩ B′
Note For more details, see operations on sets.
Probability—
Theoretical (Classical) Approach
If there are n equally likely outcomes associated with a random
experiment and m of them are favourable to an event A, then the
probability of happening or occurrence of A, denoted by P ( A), is given by
m Number of favourable outcomes to A
P ( A) = =
n Total number of possible outcomes
Axiomatic Approach
Let S = { w1 , w2 , w3 , ... wn } be a sample space, then according to
axiomatic approach we have the following
(i) 0 ≤ P ( wi ) ≤ 1 for each wi ∈ S
(ii) P ( w1 ) + P ( w2 ) + ... + P ( wn ) = 1
(iii) For any event A, P ( A) = ΣP ( wi ), wi ∈ A.
Note
l Theoretical approach is valid only when the outcomes are equally likely and
number of total outcomes is known.
l P(sure event) = P(S ) = 1 and P(impossible event) = P( φ ) = 0
