today: definition of probability. random sampling. next: asv 1
TRANSCRIPT
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