Probability

Samir K Mahajan, Ph.D. , UGC NET

 Random Experiment and Trial

Random Experiment

An experiment in which all the possible outcomes are known in advance but we
cannot predict as to which of them will occur when we perform the experiment,

e.g. Experiment of tossing a coin is random experiment as the possible outcomes

head and tail are known in advance but which one will turn up is not known.

Similarly, ‘Throwing a die’ and ‘Drawing a card from a well shuffled pack of 52
playing cards ‘are the examples of random experiment.
Trial
Performing an experiment is called trial, e.g.
(i) Tossing a coin is a trial.
(ii) Throwing a die is a trial.
 SAMPLE SPACE, SAMPLE POINT AND EVENT

Sample Space
Set of all possible outcomes of a random experiment is known as sample space.

Let S be sample space . Then total number of elements in the sample space is known as size of the
sample space and is denoted by n(S), e.g.

(i) If we toss a coin then the sample space is

S = {H, T}, where H and T denote head and tail respectively and n(S) = 2.

(ii) If a die is thrown, then the sample space is S = {1, 2, 3, 4, 5, 6} and n(S) = 6.

Note: A die has six faces numbered 1, 2, 3, 4, 5, 6

(iii) If a coin and a die are thrown simultaneously,

then the sample space is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} and n(S) = 12.

where H1 denotes that the coin shows head and die shows 1 etc.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Sample Point
Each outcome of an experiment is visualised as a sample point in the sample space. e.g.

(i) If a coin is tossed then getting head or tail is a sample point.

(ii) If a die is thrown twice, then getting (1, 1) or (1, 2) or (1, 3) or…or (6, 6) is a sample point.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Event
Set of one or more possible outcomes of an experiment constitutes what is known as event. Thus, an event
can be defined as a subset of the sample space, e.g.

i) In a die throwing experiment, event of getting a number less than 5 is the set {1, 2, 3, 4}, which refers
to the combination of 4 outcomes and is a sub-set of the sample space
= {1, 2, 3, 4, 5, 6}.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

More about Event

Occurrence of an event
The event E of a sample space S is said to have occurred if the outcome of the experiment is such that  E. If the
outcome is such that  E, we say that the event E has not occurred.

Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.

Now suppose that we are interested in those outcomes which correspond to the occurrence of exactly one head.

We find that HT and TH are the only elements of S corresponding to the occurrence of this happening (event).

These two elements form the set E = { HT, TH}

We know that the set E is a subset of the sample space S .

 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

More about Event contd.

Similarly, we find the following correspondence between events and subsets of S.
In tossing two coins at a time , the sample space
S = {(H, T) , (H, T)}
= {HH, HT, TH, TT}

Description of events Corresponding subset of ‘S’

Number of tails is exactly 2 A = {TT}

Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S = {HH, HT, TH, TT}
Number of tails is more than two f={ }

The above discussion suggests that a subset of sample space is associated with an event and an event is associated
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Types of events : Events can be classified into various types on the basis of
the elements they have.

❑ Sure events and Impossible events

❑ Simple Event
❑ Compound Event
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Sure events and Impossible events

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

To understand these let us consider the experiment of rolling a die. The associated sample space is
S = {1, 2, 3, 4, 5, 6}

Let E be the event “ the number appears on the die is a multiple of 7”. Clearly no outcome satisfies the condition
given in the event, i.e., no element of the sample space ensures the occurrence of the event E.
That is , the event E = f and E is an impossible event.

Now let us take up another event F “the number turns up is odd or even”. Here , all outcomes of the experiment
ensure the occurrence of the event F.
i.e. F = {1, 2, 3, 4, 5, 6,} = S, i.e.,
Thus, the event F = S is a sure event.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Simple Event If an event E has only one sample point of a sample space, it is called a simple (or elementary) event.

In a sample space containing n distinct elements, there are exactly n simple events. For 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}.

 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Compound Event If an event has more than one sample point, it is called a Compound event.

For 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.

S= { (H, T), (H, T), (H,T)}

= { (HH, HT, TH, TT) (H, T) }
= { (HHH, HTH, THH, TTH, HHT, HTT, THT, TTT}

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}
Each of the above subsets contain more than one sample point, hence they are all compound events.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

Algebra of events: In set theory, there are different ways of combining two or more sets, viz, union, intersection,
difference, complement of a set etc. Like-wise we can combine two or more events by using the analogous set
notations.

Let A, B, C be events associated with an experiment whose sample space is S.

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’ .

Clearly for the outcome HTT, the event A has not occurred. But we may say that the event ‘not A’ has occurred.
Thus, with every outcome which is not in A, we say that ‘not A’ occurs.

Thus the complementary event ‘not A’ to the event A is

A = {HHH, HTT, THT, TTH, TTT}

or A = {w : w  S and w A} = S – A.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

The Event ‘A or B’ : Recall that union of two sets A and B denoted by A  B contains all those elements which
are either in A or in B or in both.

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
= {w : w  A or w  B}
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

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´
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.

The Event ‘A and B’ : We know that intersection of two sets A  B is the set of those elements which are common to
both A and B. i.e., which belong to both ‘A and B’.

If A and B are two events, then the set A  B denotes the event ‘A and B’.
Thus, A  B = {w : w  A and w  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)}

Note that the set A  B = {(6,5), (6,6)} may represent the event ‘the score on the first
throw is six and the sum of the scores is atleast 11’.
 SAMPLE SPACE, SAMPLE POINT AND EVENT contd.
Mutually exclusive events : In the experiment of rolling a die, a sample space is S = {1, 2, 3, 4, 5, 6}.

Consider events, A ‘an odd number appears’ and B ‘an even number appears’.

Clearly the event A excludes the event B and vice versa. Thus, there is no outcome which ensures the occurrence
of events A and B simultaneously. Here
A = {1, 3, 5} and B = {2, 4, 6}

Clearly A  B = f, i.e., A and B are disjoint sets.

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.

Remark: Simple events of a sample space are always mutually exclusive.

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.
SAMPLE SPACE, SAMPLE POINT AND EVENT contd.
 EXHAUSTIVE CASES, FAVOURABLE CASES, MUTUALLY EXCLUSIVE CASES AND EQUALLY
LIKELY CASES

Exhaustive Cases
The total number of possible outcomes in a random experiment is called the exhaustive cases.

In other words, the number of elements in the sample space is known as number of exhaustive cases, e.g.
(i) If we toss a coin, then the number of exhaustive cases is 2 and the sample space in this case is {H, T}.

(ii) If we throw a die then number of exhaustive cases is 6 and the sample space in this case is {1, 2, 3, 4, 5, 6}
 EXHAUSTIVE CASES, FAVOURABLE CASES, MUTUALLY EXCLUSIVE CASES AND EQUALLY LIKELY CASES contd.

Favourable Cases
The cases which favour to the happening of an event are called favourable cases. e.g.

(i) For the event of drawing a card of spade from a pack of 52 cards, the number of favourable cases is 13.

(ii) For the event of getting an even number in throwing a die, the number of favourable cases is 3 and the event in this case
is {2, 4, 6}.
 EXHAUSTIVE CASES, FAVOURABLE CASES, MUTUALLY EXCLUSIVE CASES AND EQUALLY LIKELY CASES contd.

Mutually Exclusive Cases

Cases are said to be mutually exclusive if the happening of any one of them prevents the happening of all others in
a single experiment, e.g.

(i) In a coin tossing experiment head and tail are mutually exclusive as there cannot be simultaneous occurrence of
head and tail.
 EXHAUSTIVE CASES, FAVOURABLE CASES, MUTUALLY EXCLUSIVE CASES AND EQUALLY LIKELY CASES contd.

Equally Likely Cases

Cases are said to be equally likely if we do not have any reason to expect one in preference to others. If there is some
reason to expect one in preference to others, then the cases will not be equally likely, For example,

(i) Head and tail are equally likely in an experiment of tossing an unbiased coin. This is because if someone is expecting
say head, he/she does not have any reason as to why he/she is expecting it.

(ii) All the six faces in an experiment of throwing an unbiased die are equally likely.

Non-equally likely cases

(i) Cases of “passing” and “not passing” a candidate in a test are not equally likely. This is because a candidate has some
reason(s) to expect “passing” or “not passing” the test. If he/she prepares well for the test, he/she will pass the test and
if he/she does not prepare for the test, he/she will not pass. So, here the cases are not equally likely.

(ii) Cases of “falling a ceiling fan” and “not falling” are not equally likely. This is because, we can give some reason(s) for
not falling if the bolts and other parts are in good condition.
CLASSICAL OR MATHEMATICAL PROBABILITY
 AXIOMATIC APPROACH TO PROBABILITY

All the approaches i.e. classical approach, relative frequency approach (Statistical/Empirical
probability) and subjective approach share the same basic axioms. These axioms are fundamental to the
probability and provide us with unified approach to probability i.e. axiomatic approach to probability.

Let S be a sample space for a random experiment and A be an event which is subset of S, then P(A) is called probability
function if it satisfies the following axioms

(i) P(A) is real and P(A) >=0

(ii) P (S) = 1

(iii) If A , A , ... is any finite or infinite sequence of disjoint events (mutually exclusive events) in S, then
P(A orA or ...orA )= P(A )+P(A )+...+P(A )
AXIOMATIC APPROACH TO PROBABILITY contd.
LAWS OF PROBABILITY
CONDITIONAL PROBABILITY AND MULTIPLICATIVE LAW
CONDITIONAL PROBABILITY AND MULTIPLICATIVE LAW contd.
CONDITIONAL PROBABILITY AND MULTIPLICATIVE LAW
CONDITIONAL PROBABILITY AND MULTIPLICATIVE LAW contd.
Independent Events

Independent Events
Events are said to be independent if happening or non-happening of any one event is not affected by the
happening or non-happening of other events. For example, if a coin is tossed certain number of times, then
happening of head in any trial is not affected by any other trial i.e. all the trials are independent.

Two events A and B are independent if and only if P(B|A) = P(B) i.e. there is no relevance of giving any
information. Here, if A has already happened, even then it does not alter the probability of B.

e.g. Let A be the event of getting head in the 4th toss of a coin and B be the event of getting head in the 5th toss
of the coin.

Then the probability of getting head in the 5th toss is ½ irrespective of the case whether we know or don’t
know the outcome of 4th toss, i.e. P(B|A) = P(B).
 Independent Events contd.

Multiplicative Law for Independent Events