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: Random sampling.Infinitely many outcomes.Properties of probability.

Next: ASV 2.1-2.2Week 1:

Video: Prof. Todd Kemp, Fall 2019

Homework 0 (due Friday, January 8)

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Combinatorics* selecting K objects from among n ,

with replacement :# ways = n

k

* selecting K objects from among n , without replacement ;order matters :

# ways =ninth (Ksn)* selecting K objects from among n , without replacement ;

order doesn't matter :

# ways = ( K) = nlh-D-jg-h-KD-nl.nu:= He)

samplmgwithreplabmentlorde-rdsntmatte.tlIn urn contains to balls : → bi

,bzbgbybgF- -9 -

••••••••J 2 blue bgbzbgbg bio• 3 yellow

s red

Problem : 3 balls are chosen without replacement .Pl 2 yellow ,

I red )

D= { {bi ,b,b,} -

- bi#bj if #j) #r= ( Y)Idr mattersA-- { 2am yellow, l red} #A- If ) -K)

i.MA) -- Ifl-12) = = gt-12.SK

What if #r-

- A ?

1.3JThen we need a different notion of uniform .

E.g .A random real number is chosen in cell .

(a) What is the probability it is 30.7 ?

(b) What is the probability it is = 's ?-must define !

( as Picazo)(S,I,IP) =

i - e.7=0.3

[ e!!, t Tp ( ca,bi) is b-a

.Cbs Pll's . El)

BC = I-I -- o .""

pl Ceos) UI 's,end)4€

, spice, o.O , + Pll'sodd) Iff

"

" "

i :÷:÷:÷:÷÷÷÷25cm in diameter

.

A red disk in the center is

5 cm in diameter .

Given that you hit the target (randomly) , what are thechances of hitting the blue disk ? The red disk ?

A = targetg s { subsets

that have "area "}

PIA) =Arendt)

IP ( bullseye ) = 1%

Area cry

Impositions µEg . A fair coin is tossed 5 times.What is the

probability that at least 3 tosses come uptails ?

As { at least 3 tails} = Az UA40 As

A { exactly k tails)PLA) = MAD tplfhy)

tPHD← Blast Iss) Is.

T f f a f f µ :# configurations

p t t t -

- H)MB) -- (5) Is MAD -- H) - Is .[

.; pas

-

- Es list kilt KD - Ish " "DIII,

Eg .

A fair die is rolled 4 times.

What is the probabilityof at least one double ?A = l some number comes up

at least two times}

Ans f k comes upat least two times}

A = AN AmboAyo Asu As not disjointAf = In comes up exactly m times ) zillions 30

Ai ? Ain Apu Apu Apu Apscenarios

.

:

Question : Are all these events disjoint ? NotA A

'

plAT = 6.fi#3--fg④is Mrs -- P'?! '

'

p'Puff?i. payers. -- Fs

Sometimes, you can't avoid lack of disjoint ness se easily .

You have to take intersections into account.

Notation : AaB = { all outcomes in both A and B)11

HB

④µBAOB = AB

'

u AB u A'

B ← disjoint

Pl AUB) = MA) t PIB) - PlAaB)

prihcipleefindusion-xcusiwt.hrprobability of a union can be computed by addingthe probabilities, then subtracting off the intersection is )ever counted . If you have more sets

, you have to keepgoing and re - add back in pieces that you over - subtracted,ok . C

P 1A uBuc)

"ftp..sk#i:k.m.cst PlABC)

Eg .

20% of the population own eats .25% of the population own dogs .

5% of the population own both .What is the probability that a random person owns neither ?

C D

'

¥ ÷:naif

= I - pl CUD)= I - ( PIC) t PID)

- PKDD= I - ( o .2 to. 25 - ees)= o

-G .

MonotonicityIf A s B

④them: Ig:B.

d.i.int union

Eg. 9g of your friends like the xiao long bae at Din Tai Fung .of your friends like the xiao long bae at Shanghai Saloon .

What is the smallest possible proportion of your friendswho like the xiao long bae at both restaurants ?