Today: Definition of Probability. Random sampling. Next: ASV 1

MATH 180A: Introduction to ProbabilityLecture B00 (Nemish)

www.math.ucsd.edu/~ynemish/teaching/180a

Lecture A00 (Au)www.math.ucsd.edu/~bau/w21.180a

Today: Definition of Probability.Random sampling.

Next: ASV 1.2Week 1:

Video: Prof. Todd Kemp, Fall 2019

Homework 0 (due Friday, January 8)

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The world around us is fundamentally random .

* Hoos - games of chance* Does - quantum theory* Uses - finance Insurance* uses - chemical reactions inside enrolls* zoos - complex networks ; machine learning

The modern rigorous foundation of probability theory goesback to 1933 Kolmogorov

Ingredientssample Space r -- the set of possible outcomes in anexperiment . { HH, HT, TH , TT}Events I = collections of outcomes . E -- {atleast l H}

Probability Measure p : I → Cgl ]IHH HITA)

kolmogoro.is/txioms- IPIEUT) =3

Ii ) For any event ASF,

o SPCA) St.

" I

PIEJHPLT)Lii ) pls) = t Pl = o

{ + §/

Liii ) If Agha,As,.. . are disjoint events , then g

PLA,uAwAzo - - - ) = PIA,)tPlAs)tPlAz)t - - -

Eg. A fair die . 52=11,334,963 PHD - Phat - - - =P

E. seven,O -- odd E - {34,6) 0=43,53

Ts divisible by 3 T={ 3,6} PIE) -- f- s 's

PIEOT)

=p { 2,3 ,4,63EVO 'S PIT)=f= }

.

"

1%34.mg?tPY.BtlPH3EuT-- 1334,031-

Probability space l r,F

, PfT tsample space events probability measure

Determined by the experiment being!

Key property : A Ai,Az,As,. . . are disjoint ( Aif . ¢ )-

C- 5-

then PLAN AN Bo - - - J -- § PIA;)Important special case : if its so then

I = { wi,wz,wz,

- - --

,Wn}

the singleton sets good , { wig , . . .

,fun} are disjoint

i. PIA ) = 193019230 - -- Har's PLA) - fi, pKa;DA = { ai , Az, - -

yar) -

uniformprobabilitymeasw-eampngly.LIf r is finite,the uniform probability measure

is defined by :%

For each wer,P = #I

#A⇒ Forany event A

,MA) = #q

This means calculating probabilities in suchmodels is tantamount to County .

Eg .

A fair die is cast 2 times.

What is the probability thatthe sum is 4 ?D= { ( i. j) : Isi, jst } #1=36

ismfstqj {"'3) , 1323, 13, D} #A =

3) MA? IIZ

E 8.3%

Eg . A fair coin is tossed 3 times .

A -- { at least two tails}

B :{ exactly two tails}

e. = E AHH,HHT

,-- -

,TTT} #r=g

= { lijjgh ) : i.j,REST, H} )

A = { TTH,THT

,

HTT,

TTT } PIA)=#Ag=4f='s

B = { TTH, THI HTT} PIB) -- 3g .

ttlNkPSHARE

There are 10 people on a committee .

How many different ways are there to selecta subcommittee of 4 people ?

(a) 10.10 - lo - to = 104 = 10,000

(b) 10 . 9 . 8 - 7 = 5,040

( 140) = 10-4-8-7 = 210

(d) 10¥.

= 151,200

Combinatorics

.••..:.③[email protected] replacement , orr:{ lb , bid :

f* without replacemeh-tks.ru )

is bish } > They are lined up fbi #b; if Hj

= shrink* in the order they came out, or

How many ways ?* disregard gag? g

* with replacement : n"

* without :{& ordered : h - Ch -D - th-2) - - - (n-ka)

net : nlh-blhm.tt#h-ktD--n.Y-n.ny.= ( L)

samplingwithR-pam.nlToss a fair can n times ; record a statistic observing

* H Is.

HT.

Eg . n = 10,

IP { odd rolls are all H}.

" D

D= { ( Ci,s,G,-→ Go) : tj g. El H ,T}}

He = 210

A = { c,= Czs Cs

= Cps Cg = H)-

- SCH,*, Hot, H, *Hot, H,HA = 25

PLA) = 2¥ -- Es ' Ii

samplmgwitheutreplacementlordermatte.gsth:: aristate!:D .}:"esn.in:*:* .

replacement ) , and lined up in order .what is the probability that the firsttwo are 13,6 ) ?

S = { (bi, bgbz) : is bit 6 , bit by bubs, b , the}*r = 6-54

= 120pls) - Ya

A = { 13,6, *I } HA -- 4=

1,

-30

-

C- { 1,34, s}