Probability

History
• History of Probability goes back in mid sixteenth
century around 1654.
• Chevalier de Méré, a French nobleman with an
interest in gaming and gambling questions, called
Pascal’s and Fermat’s attention to an apparent
contradiction concerning a popular dice game.

The Problem of Points
• Question posed by a gambler, Chevalier De Mere and then discussed by Pascal
and Fermat.
– There are many versions of this problem, appearing in print as early as 1494
and discussed earlier by Cardano and Tartaglia, among others.
• Two players of equal skill play a game with an ultimate monetary prize. The first
to win a fixed number of rounds wins everything.
• How should the stakes be divided if the game is interrupted after several rounds,
but before either player has won the required number?

Basic terms
• Random Experiment
• Sample Space
• Event
• Complimentary event
• Union of events
• Interaction of events
• Difference of events
• Mutually exclusive events
• Exhaustive events
• Equally likely events
• Independent
• Favourable events

Random Experiment
An experiment which can result in any one of several possible
outcomes is called a random experiment.
Characteristics of a random experiment
• The experiment results in any one of the several outcomes
• All possible outcomes of the experiment can be described in
advance but cannot be known in advance
• The experiment can be repeated under same conditions

Sample space
A set of all possible outcomes of a random experiment is called a
sample space.
E.g. Consider an experiment of throwing a dice,
then the sample space, S = { 1,2,3,4,5,6}.
E.g. Consider an experiment of result of a student in an exam,
then the sample space, S = {Pass, Fail}
E.g. Consider an experiment of throwing two coins,
then the sample space, S = {HH, HT, TH,TT}

Event
The result of a random experiment is called an event. or
A subset of a sample space is called an event.
E.g. If S = {1,2,3,4,5,6} then event A of getting an even number is
A = {2, 4, 6}
E.g. If S = {HH,HT,TH,TT} then event B of getting atleast one tail is
B = {HT,TH,TT}
E.g. If S= {1,2,3…100} the event C of getting a number divisible by 3 is
C = {3,6,9,12,…99}

Complementary event
The set of all the sample points of sample space which do not
belong to A, is called the complement of an event A.
E.g. If S = {1,2,3,4,5,6} and A = {2,4,6} then A𝑐 = {1,3,5}.

Union of events
The set of all sample points belonging to either A or B or both is
called the union of events A and B. It is denoted by or
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A U B = {1,2,3,4,5,6,8,10}
Theevent that either A or B or both A and B occurs is known as
the union of two events A and B.
A B
.

Intersection of events
The set of all sample points belonging to A and B is called the
intersection of events A and B. It is denoted by or
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A ∩ B = {2,4,6}
The event that both A and B occurs together is known as the
intersection of two events A and B.
A B
.

Difference of events
The set of sample points belonging to A but not in B is called the
difference of events A and B. It is denoted by A - B or
If the event A occurs, event B does not occur, then it is called the
difference of events A and B.
E.g. If A= { 1,2,3,4,5,6} and B = {2,4,6,8,10} then
A − B = {1,3,5} and B – A = {8,10}

Exhaustive
If all possible outcomes of an experiment are considered, the
outcomes are called to be exhaustive.
E.g. If S= {1,2,3,4,5,6} , A= {1,2,3,5} and B = {2,4,6}
then A and B are called exhaustive as A U B = S

Mutually exclusive
If the occurrence of one event prevents the occurrence of
remaining events then events are called mutually exclusive
events. or
If events cannot occur together then they are called mutually
exclusive events.
In other words A ∩ B = Ø
E.g. A person cannot be of age 15 years and
also be voter of India.

Equally likely
If the events have equal chance of occurrence, events are called
equally likely events. or
When none of the outcome is expected to occur in preference to
another, such events are known as equally likely events.
E.g. In tossing of coin, outcomes Head or Tail are expected to occurs same
number of times.
E.g. In throwing a dice, chance of getting number 1 or 2 or 3 or 4 or 5 or 6 are
expected to be same.

Independent events
If the occurrence of one event does not depend on occurrence or
non-occurrence of other events then events are called
independent events.
E.g. In an experiment of tossing a coin and throwing a dice. Event of
getting Head on a coin does not depend on getting even number or odd
number on dice.

Independent events
E.g. Three persons say Rahul, Narendra and Akhilesh aim a target.
We know the chance of Narendra hitting the target does not depend on
whether Rahul or Akhilesh hits the target or misses the target.
E.g. Two students of HL, one of Section 2 and other of Section 4 are given
a statistics problem to solve. We know the chance of Section 2 student
solving the problem does not depend whether student of Section 4 solves
or does not solves the problem.
E.g. For a couple if husband dies can we surely say wife will also die. She
may or she may not.

Favourable cases
The sample points in the sample space that helps in the occurrence of
an event are called favourable cases.
E.g. If S = {1,2,3,4,5,6} and event A is even number
then favourable cases are {2,4,6}.
E.g. If a card is selected from a pack of 52 cards and the event B is getting an Ace
card then favourable cases { Club Ace, Heart Ace, Diamond Ace, Spade Ace}

Mathematical or Classical or
Apriori definition of Probability
If the experiment can result in 𝑛 exhaustive, mutually exclusive
and equally likely ways and if 𝑚 of them are favourable to the
occurrence of an event A, then the probability of occurrence of
an event A is denoted by P(A).
i.e. P(A) =
𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑐𝑎𝑠𝑒𝑠
=
𝑚
𝑛

Mathematical or Classical or
Apriori definition of Probability
E.g. Consider an experiment of throwing a dice. Find probability of getting
number greater than 4.
Here, sample space: S= {1,2,3,4,5,6} then favourable cases for the event A
of getting number greater than 4 are { 5,6}, so
P(A) =
𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑐𝑎𝑠𝑒𝑠
=
𝑚
𝑛
=
2
6
= 0.33

Statistical or Empirical or
Aposteriori definition of Probability
If an experiment is repeated under essentially, the same
conditions for a large number of times then the limit of the ratio
of number of times the event occurs to the total number of trails
is defined as the probability of the event.
i.e. P(A) = lim
𝑛→∞
(
𝑚
𝑛
)

Modern or Axiomatic definition of Probability
If P(A) is the real number assigned to the subset A of the sample
space S, then it is called the probability of an event A, provided
P(A) satisfies the following postulates:
Postulate 1: 0 ≤ 𝑃 𝐴 ≤ 1
Postulate 2: P(S)=1
Postulate 3: If 𝐴1, 𝐴2, 𝐴3,… is finite or infinite sequence of
mutually exclusive events i.e. subsets of S then
𝑃(𝐴1 ∪ 𝐴2 ∪ 𝐴3 ∪……) = P(𝐴1)+P(𝐴2)+P(𝐴3)+…..

Conditional Probability
The probability of occurrence of an event B, when A has occurred
is called conditional probability of B under A and it is denoted by
P(B/A).
P(B/A) =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
; P(A)≠ 0

Bayes Theorem
If an event D can occur with mutually exclusive and exhaustive events
𝐴1, 𝐴2, 𝐴3,…..𝐴 𝑛 and if the probabilities P(𝐴1), P(𝐴2), P(𝐴3),…..P(𝐴 𝑛)
and also the conditional probabilities P(D/𝐴1), P(D/𝐴2), P(D/𝐴3), …….
P(D/𝐴 𝑛) are known then the inverse probability of occurrence of an
event 𝐴𝑖 under the condition that the event D has occurred is given
by
P(𝐴𝑖/ 𝐷) =
𝑃(𝐴 𝑖∩𝐷)
𝑃(𝐷)
=
P( 𝐴 𝑖)P(D/ 𝐴 𝑖)
P( 𝐴1)P(D/ 𝐴1)+P( 𝐴2)P(D/ 𝐴2)+P( 𝐴3)P(D/ 𝐴3)+..…P( 𝐴 𝑛)P(D/ 𝐴 𝑛)

Probability

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