KALAYANI GOVERNMENT ENGINEERING COLLEGE
TOPIC:- Axioms on Probability and Baye’s
Theorem
NAME:- Gourab Mondal
ROLL NO:- 10201624055
REGISTRATION NO:- 241020110338 of 2024-25
SUBJECT:- Mathema cs (III)
PAPER CODE:- BS-M301
DEPT:- ELECTRICAL ENGINEERING
YEAR & SEM:- 2nd Year,3rd Sem
2025
KALAYANI GOVERNMENT ENGINEERING COLLEGE
TABLE OF CONTENTS
PAGE NO CONTENTS
1 Introduc on: Some basic
defini ons related to
probability.
2 Axioms of Probability: Write
the axioms
2-3 Applica ons of Probability
Axioms: Give proof of some
formula using axioms.
3-5 Condi onal Probability and
Baye’s Theorem: Give
defini ons related to
condi onal probability,
independent events; state
and prove Baye’s theorem.
5-6 Applica ons of Baye’s
Theorem: Give solved
examples related to Baye’s
theorem.
Pro6aBliit
Some DefBuattony
* Rondo Gxperimevt cE) ly out cones of & are
outcoMe
Kmowm iy TmposiBle to predlet auy
E ean be yepe ated au no. of Hmes
owte es mes
* Sample space| vew sp ace? S=
Ac S
S
* certain Evet : SCS
Events
* Simple ond composite
A13 B,= 2,4,6)
Bn H,T}
A H}
ex chwive Events :
AB=
* Mutually
S- ua,3,4,5,6}.
A256), B 3 , 5 3
* pairwlse exclaive guents
ents
A+0=s
* comptement vent
B, =A
Mondah
P:A R
TR.
Axibm 1.
PCA
PCS) = 1
AxI0m 3
mutualb ercytve
P(A,+ At Agt --)= PlA)+ P(A)t P(A:)
* Sowe Defimatiovy
. PCAS) 1
Proof:
Iy 4S =Sl=) PCDt s) = P(s)
t9s-ø]
At A
) PGA) + Pn P(6) CAAS= p)
PCAC)
1 ) pn) < 1 AeA
ospA) S 1
P(AB)
IN) p(A+ B) = P(A) + PeB)
iv)
Proof;
6
B- AB
P(B-AB)
P(AtB) PA- Ae) + p (AB) t
P(y6) + PA0)+ P(B)- PAB)
=) pCAt 3) = P(A) -
P(AB) PCBJ|
P(A)t P(B)+ cC)-
*P(At B+©)=
Conditiowal proß abiil4y
* P(AJ6) = P(as)
P(B)
* PB)A) Pn B)
PCA)
f(Ale) Ps)
P(AB) = B
- PL/A) P()
* Sudependent erents
P(AR) PCA) P(8)/
P(Rle)= p
*Paivwise gudependet tually gudepeudeut
PenB) F P)P (3)
Mutu all p(Bc) pB) pc) paiywie
frdependeut
-endet
PAGc)= PA) PB) Pcc)
* Baye's Theorem
Statemeut: Let A,,A)-~-) Ay Be mutuall
exelugle and exthaytte evemts nyo Let B
Be an otheY event Fo) conec ted
PCB) #o)
P(B)
P(A;s)- P(Ai) P (3/Ai) ,lz y 2,
)
n
n
P(A)Pe/A;)
Proof?
) (A t A2+ -+ My) B SB
t Au g) PB)
P (A Bt A2 8+
(Ai) (B)
(Ai A)(9 B)
) PtA, B)+ P(A73)t - t PCAn B)pB)
P AjB) PB)
P(Ai) p(6/Ai) = P(3)
P(AiB) P(Ai): P(B|ai)
pCA/B) = P(Ai) P (G/Ai)
a chance for a
Suppose there is
*Ex: conytrueted howe to e o l l a p s e
vewdy fauty oY not.
WeaHer tte design is sigu s fautJ
Tae ctane at t e de
ta at tee thowye
is |07. the chauee faulty is 95.
he desl n ts
fs
Collapses ie '% Tt is
Is Seey
t is. 45
a d otercufs e Colla ed.
tht a hoe to fault
proBal1 taat t Is due
is fulty
A’ peign fauty
D-yan is mot
A e ollap Sey
B toy e
) PCA 2)
f(B/An)= le|# 2) 9
20
AlB) = P(A) P (B)A)
p(
P(A)P(B/A)t pl) P(B/)
X19
19 19
19+81
20 20
Refevenc ey : yoUTuBE LECTURE OF
DR.PRASANTA BAsAK stR
(DEPT- HoD oF MATAEMA Tes,
(KaEc))
THA Nk YoU
