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MT2504 Combinatorics and Probability Chapter 1
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Sets , counting and elementary probabilityCHAPTER 1
Definition :
sets A ,
B are
disjoint if A⑧ =
0 . Sets Ai ,
- - . ,An are ( pairwise)
⑦ Ai
)
disjoint if
,
Aj disjoint (
i #
j)
↳ for all
a g
Theorem : 1.5
Let A O be finite and sets then
, disjoint ,
IAU 81 =
IAI t
181
Proof :
het
A =
{ a
, .az ,
-
- -
,
am } for some M n EIN
,
B =
{ b. ,
bz
,
- - .
,
bn }
AUB - { a
, .ae . . . .
,
am
,
bi
,
bz ,
-
-
-
,
bn }
↳ Because ( sets disjoint) ( Au ol IA It 181
repetition are hence Mtn
'
no
-
: - -
,
theorem : 1.8
Az , . . .
,
An pairwise disjoint finite sets .
I A. U .
. .
VANI =
IA It ,
-
- - t
tant =
Eh ,
tail
Proof -
.
induction ( already done n 2) → Theorem 's
By
-
on n
-
.
suppose true for n > 2 .
Let Az ,
-
. .
,
Anti disjoint finite sets be given .
Set A =
Aau . . -
VAN
since An Ant , =
( Ain Anti) = ¢ since all Ai are pairwise disjoint .
A and Ant ,
'
are
disjoint
-
'
By inductive (a )
hypothesis z
-
:
-
IAU And =
( Alt ( Ant il
REMEMBER :
IAI =
I Aa U . .
-
U Ant = IA , It - - - t ( Anl
SO
,
I Azu . . . U Ant I ,
= ( Alt l Anti ) =
( Att - - -
t
tant I ,
, Observations :
Let A. 8 be sets
IAI 30 ( ( Al o ⇐>
A 0 )
-
• -
=
•• A EB IAI ← 101 B = AUB LA → A ,
8 disjoint
•
Az ,
.
. .
,
An disjoint if Ail ,
=
is ,
I Ail
Definition ( 2.1 )
let r be a finite sample space :
A probability IP : → IR
assigns every event a number s - t :
→ Collection of all subsets of R
AXIOM 1 : ( non -
negativity ) VA Os PLA ) EI
AXIOM 2 : ( honesty) ( pls) -
-
1
AXIOM 3 : ( additivity) Az ,
. .
.
,
An disjoint PCA , U . . .
Van ) =
i. MAI )
Definition X. 3)
If IP is a
probability on r
,
the function f : r → IR
,
f- ( w ) =
BLEW } ) is
the
probability mass function of IP .
Lemma ( 2.4 )
het r =
{ wi }? , ,
IP
probability so that each outcome is
equally likely .
Then
,
Ant IAI
'
✓ events A Er IRA ) =
=
,
Irl
proof :
flw ) t Er To
compute p
:
w
p
-
-
.
Is D= It )
>
1 :
Ip ( r) =
,
flwi ) =
rip ,
{ wi } IP ( { wi } -
-
Hwi)
,
-
-
Honesty Additivity
so
p
-
-
ht
Now let A er be
given ,
Ant
'
IPCA ) =
Efcw) =
Http =
WEA
↳ sum over elements of A
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