Class notes
MT2504 Combinatorics and Probability Chapter 1
- Course
- Institution
Handwritten revision notes
[Show more]
Seller
Follow
WinterBerry
Content preview
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller WinterBerry. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.60. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
79650 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now