Probability

Definition of Probability:
Let S bea sample space of a
random experiment.
Mathenatical or Classical or
Let S has finite number of pointsApriori
and all Definition:
points shas cqual Ichance of occurrence.
Let A be any event associated with S,
then protbability ol As
PÍA)A)_ No. of pts in A
#(S) No. of pts in S
Statistical or Empirical or Posteriori Definition: condition,
If an essentially identical
experiment is performed repeatedly n times under
and na times event A has occurred then probability ofA is
P(A)="A. as n’ 0 i.e. for large numbers of trials n

Axiomatic Definition (Kolmogorove Definition): associates a real

number to the
which
Let Shas finite number of points aand Pbe a function
set of events of S such that
i. P(A)20 for any event A ofS
i. P(S) =1
ii. If A& B are mutually exclusive events i.c. AlB=
then P(AUB)= P(A) +P(B)
Results:
I. P(0) =0

2. 0sP(A)sl forany event A of S

3. P(
4. Addition Law: P(AUB)= P(A)+P(B)-P(ANB)
s. P(A, UA,UA,) =P(A|)+ P(A,)+P(A;)-P(A,NA;)-P(A,NA;)
-P(A;NA,) +P(A,NA,nA3)
Exercise

Ex. Acommittee consists of 9 students, 2 of which are from 1" year, 3 from 2nd and 4 from 3d
year. Three students are to be removed at random. What is the chance that
i. The three belong to the different classes. i. Two belong to the same class and third to the
different class. i. The three belong to the same class.
Conditional Probability:
Probability of an event A iven that event B has already occurred is known as
probability of A with respect to conditional
B: denoted by P(A/B).
P(A/B)- P(ANB)
P(B)
Similarly P(B/A)P(ANB)
P(A)
Multiplication Rule:
P(ANB)=P(A/B) P(B)= P(B/A) P(A)
Independent Events:
IWo evecnts A
occurrence of theandother. are said to be independent
B
if thbe occurrence of any one
does not aftcct the
ie. P(A/B) = P(A) and
Result:
P(B/A)- P(B)
I. Two events A and B are
independent ifr P(ANB)=P(A) P(B)
Similarly Aj,Az,..., An are independent it
P(A,MA,n...nA,) =P(A) P(Az)... P(An)
2. Ifevents A and B
are independent then
i) A and B are
i) A and B are
independent
independent
Excrcise
1. When a die is tossed,
an odd number has
what is the probability that the result is less than 4 when it is
appeared? known that
2. A boN contains 20 defcctive and 80 non
is the probability that defective items. Two items are chosen at random,
both are defective when
i. 2nd item is chosen what
without
ii. 2d item is chosen after replacing the item.
replacing back the item
Theorem of Total probability:
let By.B,..B, be a colletion of mutually exclsive and exbaustive cvents of a
smple space Sic. B,NB,¢vizjand B,UB,U..UB, S.
Then for any event AofS,
P(A)= P(A/B) P(B)+P(A/B,) P(B,) +.. P(A/B,) P(B)
Baye's Theorem:
Let B.B,.... B, beacollection of mutually cxclusive and exhaustive events of a
samplespace S i.e. B, NB, -Vizi and B, UB, U.UB, -S.
Then for any event A of S,

P(B,/A)= P(AB,) P(B,)

P(A)
P(A/B,) P(B,)
P(AB)P(B)+P(A/B,) P(B,) +.. +P(A/B) P(B)
Ex. There the two brands of regulators available in the market. Aperson may buy a regulator ol
brand Xin 75°% and that ofY is 25%. 1r959% of brand Xund 80% him
of brund Yperlorm according to
work AcCording to tlhe
regulator bought by
the specitication. Wht iis the probability that a workine
specitication? If the regulator t by him is according to the specilication, what is tle
probability that it is of brand Y?

Es. Froma vesscl containing 3 white and § black balls, 4 balls aretransferred inteo an empty vessel.
From this vessel a ball is drawn and found to be white, what is the probubilitythat out of four balls
transferred 3 are white and I is black? Ans; 0.14
Randonm Variable:
Let S be the
sample space of an
A
function X:S’R which assignscxperiment.
atreal numbers s X(s)to every seS is caled a
Random variable (r.v.).
The set Ryx =(X(s)/ seS) is
Nole: I1 outcomesof an
called the Range space of X.
as r.v, and Rx experiment are already numerical yalues, then we take X(S)=S
=S.
Def
1. If Rx is
finite or countably
2. 1f Ry is
an infinite then X is called a Discrete r.v.
interval then X is called a
Continuous r.v.
Probability
Let X be a Distribution Diserete
of r.v.:
discrete r.v. defined on a
ie. Rx ={x1,N2,..xË,., sample space S and takes values
A function
p:R, ’R
N1,X>,.Nj
is
known as
iff p(x;)20 Vi=1,2,.. Probability mass function (p.m.t.) of the r.v. X
Xp(x)=1
The r.v. X along with
p.m.f p
X :
associated with it
N..
P(*) : p(x) p(x2) p(x3).
is known as a
Probability of distribution the r.v. X
Note: Lct BcRx be an event associated with r.v. X,
then
P(B)- xeB>P{X= x}= p(*) KEB

Distribution function for Disereter.v.:

Let X be a discrete r.v. with
probability distribution
X :
X3...
p(x) : p(x) P(x2) p(*3)..
Then a function F:(-0, o)’R defined as
F(x) = P{Xs x}= P{X=x}= p{x} is known as Cumulative distribution
X;Sx
function (c.d.f) or simply distribution function (d.f.).
 rN takes finite numbers of values Nj,X),.... Nn then
0 -0<X<

P() XSx<X2
F()= P(x)+p(xz)

P(N)+p(x;)+...tp(N, )=l XSx<o

Or

F(N) : 0 p() p(x)+p(x2) ... p(x1) +P(x2)+... t p(x,)=l

Results: Let F(x) be a d.f. of ar.v. X (discrete )
1. P()=P{x-x}=F(;)-F{,-1)
2. P{a<Xsb} = F(b)-F(a)
3. P{asXsb} = P{X= a} +P{a< Xs b}
4. P{a<X<b} = P{a <Xsb} - P{X= b}
s. P{asX<b} =P(X= a} +P{a<Xsb)-P{X = b)

Exercise

1. For the r.v. Xwhich denotes the sum of the outcomes when a pair of fair dice are rolled
i. obtain the p.m.f. and probability distribution of X.
ii. ind the dr
iii. find the probability that sum of outcome is an odd number.
iv. find the probability that sum of outcomes lies betwcen 3 and 9?
2. A r.v. X has the
distrtouton
6 7
p(x): 0 k 2k 2k 3k k? 2k? 7k+k
Find i) k ii) evaluate P(X<6). P(Xz6). P(1< X<5) i) find the minimum value of aso
that P(X sa)> ii) Find the distribution function of x.
A r.v. Xhas the distribution
X: 0 3 4 6

p(x): k 3k Sk 7k 9k 1k 13k
Find i) k i) P(X<4), iii) P(3<X<6)