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  3. Stellenbosch University (SUN)
  4.  
  5. Probability Theory And Stats 114 (PTS114)

Summary

Summary Probability Theory and Statistics 114 Summaries

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  • Course
  • Probability Theory And Stats 114 (PTS114)
  • Institution
  • Stellenbosch University (SUN)
  • Book
  • A First Course in Probability

A concise neatly digitally written summary for probability theory and statistics 114. All important concepts and examples for you to ace your exam for probability theory and statistics 114. * Please note that the pages will be displayed as full A4 pages when downloaded (not a small version like in...

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  • No
  • Chapter 1 to 5
  • June 11, 2022
  • 28
  • 2021/2022
  • Summary

Subjects

  • probs and stats
  • probability theory
  • probability theory and statistics
  • probability theory and statistics 114
  • probability theory and statistics summaries
  • probability theory and statistics 114 summaries
  • p

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Chapter 1: combinatorial analysis




elements

" "" " "
in order / ordered

arrangements ( basically order of elements matter )
,



0 e.g. ABC ≠ BAC ↓
considered
n
distinguishable objects
as diff
.
sets




Total
☐
same elements in diff
sets are the same
same




"
connects to ABC = BAC =
BCA
the Binomial
theorem > distinct objects




• Proof of the basic principle of counting ,
.
m outcomes
.




µ
2 experiments are
performed Together
•

: mxn
→

•



Experiment 1 has M possible outcomes % i




2)
,
n outcomes

possible
3
Experiment 2 has n outcomes
•

a


• If
exp .
I can result in
any 1 Of m
possible outcomes for each possible outcome of
exp . I and there are n

possible outcomes, then together there are mn possible outcomes of the 2 experiments

, ( 1,1 ) ( 1,2 ) . . .
( 1 n )
, , ,




( 2 , 1) 12 2) . .
(2 n )
,
.

, , ,



:



Cm
,
1) ,
( M 2)
, ,
- . .
(M
,
N ) ] Total : mxn



i. set of possible outcomes consists of m rows
, containing n elements




•
Permutations :

Number of permutations of n distinct objects :




n ( n -

1) ( n -

2) . . .
2- 1 .
= n !




With different groups of groups of arrangements / permutations :

If the groups can be switched around : XM !
,
where m is the no .
Of places
4 !
e. g. 4-1 . .
3-1 .
✗ 2! .
1 ! ✗




Number of of objects where identical
permutations n
,
n, ,
nz are :




formula of
If we
just use normal
,
we are
going to
get repetitions arrangements .




e.
g. BOB

: we need the number of distinguishable / different permutations :
Pn .




Pn
n.im?!...nr,. number of permutations of n
objects where hi nz nr are identical
=
.
. .

, ,




e. g. • PEPPER problem
• 10
competitors where
just nationalities are listed
•
Different arrangements of objects of identical colours

•
Handshakes

•
Grid problem
6!
• A BEFORE B with A B C D E F }
2 !
^
A B

, combinations


choosing groups ( combinations ) Of size r
,
from a collection of n objects Groups : not


ordered !
-
not ordered : ABC =
BAC
;
because groups cannot be counted as
permutations


Number of objects from collection of n objects where order doesn't matter:
ways to
pick r a
,




n ( n -
1) ( n -




r!
2) . . .
(n -
r -11 ) =




r
n


! (n
!
-
r) !
=

(Y) ) n choose r




examples :




•
choose a Comm of 3 from 20 ¥
•
men and women comm : (E) (F) = total outcomes



> 2 men refuse to serve together (E) (( E ) ✗

-
-


(mm : )(I÷mf ) )
" "

normal where
men group
group
2
feuding men

serve together


Linear configurations
of which defective
Example of n anntenas
,
m are .




How many linear configurations (
permutations) where no 2 defects are next to each other ?
,




① up then Basically find Of
Line
working antennae : n m no
-
.


,
ni na
ni
defective antennas
}

places for
in
^
T
^ th tht out )
( throw working ones



spaces for
② defective
There are In -
m -11 )
possible m antennae .




③ of these
At most 1 defective anntenae can
go in each spaces
( )
n - m + '


④
NB
SO we choose M leg .
2) of the ( n
-

m -11 ) ( e. g. s )
spaces : : .
m possibilities
e. g. (E)
Of to
no . ways
< ↳ from 5
this is the no .
Of possible choose 2 places

orderings where there is at least

I
working antenna between 2 defective ones

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