J.M. Villalobos

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Math 150 Lecture Notes Date

§4.1 Probability Basics

• Probability Experiment

A chance process that leads to well-defined results calledoutcomes.

- Roll a die

• Sample Space

The set of all possible outcomes

- 1,2,3,4,5,6

• Outcome

The result of a single trial of a probability experiment.

- Getting a ’4’

• Event

A set of outcome(s) of a probability experiment.

- Getting an even number

1

J.M. Villalobos

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� 2016

Probability types

Empirical Probability

� P (E) =frequency of E

Total number of trials

Subjective Probability

Classical Probability

� Equally likely outcomes.

� P (E) =n(E)

n(S).

2

←

pratfall assFlip a coin

1000 times

Count # of tails

111T 1 .-

- -

=485

. 0

:Bias )

Personal Experience

=# of outcomes in E

÷ al # of outcomes

J.M. Villalobos

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� 2016

Ex:

• Probability Experiment

A family decides to have three children.

• Sample Space

• Event

E = The family has two girls and one boy

A = The family has no boys

3

Sample Spaceb

bbb1¥ €

y bbyb bgb

/ b byg

-b-st.qq.gg/8aoias

-3

\ -b \ g

as

\g f:

PC E) =3-

8

Plno boys )=P( ggy ) = gt

J.M. Villalobos

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Ex:

• Probability Experiment

Nacho takes a test with three questions. Two questions are multi-ple choice (5 choices) and one True/False question.

• Sample Space

• Event

E = Nacho gets all three questions correct.

A = Nacho gets all three questions incorrect.

4

3rd QuestionS

←Cho '

⇐

c

1¥ 4 choices 2nd 3-choices

kwc

sample Spacec < w

.

-

w < w CCCW

a C C w

est → # < w

cma

:

¥g:| "

f- w-wt#- wTwfwe:we

:- W -

C g :A

we:

W

g :

PCALL 3 correct ) = I24

PC A )=p( www ) = 6-24

J.M. Villalobos

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� 2016

Probability rules

� 0 P (E) 1

� P (E) = 0

� P (E) = 1

Complement of an Event

� P (E) = 1� P (E)

� E = At least one

5

championship

)

( Clipperwinning

a

⇒ E will never happen

⇒E will always happen

E E ( not E)

Atleast

one

÷,

3,

4,

5,

6,

8

P

E = none zero we)

( w

{¥( o kids

÷at least one buy

PC E) = 1 - P( no boys )

=1 -PC ALL 10 are girls )

= 1 - 1Luzy

=10231024

J.M. Villalobos

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§4.2 Addition Rule

P (A [ B) = P (A) + P (B)� P (A \B)

Mutually Exclusive Events

Ex: A math class has 60 students. 35 students like chocolate, 25students like strawberries and 15 students like both. Find the followingprobabilities.

6

And

←OR k

Aand B

happenat

(the same

time )

PC An B) = 0A B

Honest -←

politicians

C Sc ) P( E) = 2560

d)PC 5) = 3560

15 10

e)P ( Ins ) =D

20

60f)PCC ^5 ) = 2-0 1560

a) PKCUS ) = PC c) + PCs ) - P ( Cns )

=3÷o+¥ - t÷=4÷b) PCC u 5) = Pcc ) t PC 5) P ( Cns )

= To + E -Eg=5E

J.M. Villalobos

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� 2016

Ex: The following table summarizes the car manufacturer of 300ECC-CC students.

- American Japanese German

Male 20 100 30Female 15 95 40

(a) What is the probability that a randomly selected student owns

an American Car?

(b) What is the probability that a randomly selected student is fe-

male or owns a German Car?

(c) What is the probability that a randomly selected student is Fe-

male or the student does NOT owns an American Car?

(d) What is the probability that the student owns a Japanese Caror an American car?

7

1 50

150

3 5 1 95 70 130-0

PC A ) = 35300

•P ( F U G ) = P ( F ) t P ( G ) - P ( F n a )

* =

Toto + Foto- goto = soooo

•

P ( F U At ) = P ( F ) + Pc A- ) - P ( F n TA )

:= goto + 23¥ - BE

=mix::#

P ( J U A) = P ( J ) t P ( A) - P ( J n A)

=

get + To - fo = Soto

J.M. Villalobos

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§4.3 Independence

Two events A and B are independent (A?B) if the occurrence of A

does not a↵ect the probability of B occurring.

If A?B then P (A \ B) = P (A)P (B)

8

E

¥ PCL )= 0.4, p( D) = 0.3

,PCLN D) = 0.15

?

PCLND ) IPCLJPCD )

←not

Independent.

0.15 I0.4 ) ( 0.3 )

0.15¥ 0.12 ⇒ LI D

{¥ West: 1st : 4 choices,

2nd : 3 choices 3rd : 2 choice ,

P( ALL 3 Questions Wrong )

=P ( w, nwznw } )

=P ( w , )P( Wc ) PCW } )=

3g . § . tz

= 6-24

k¥ PC Face Card )= I , PC Red Card ) = 26A 52

B 52

PC An B) =6

PC An B) I PCA > PCB ) 52

7 6.

Es÷÷s±is÷=s÷Hit⇒ ATB

✓ ( Independent )

J.M. Villalobos

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� 2016

Ex: Mickey Rats goes to dollar store and buys two alarms. Theprobability that the alarms work any day is 90%. What is the proba-bility that:

(a) both alarms will work tomorrow?

(b) he will wake up next Saturday?

Ex: According a recent survey 70% of Angelinos like chocolate. Ifyou randomly pick 6 Angelinos, what is the probability that at leastone of them likes chocolate?

9

PCW , )= 0.9,

P( we )= 0.9

PCF , )=O .1

, p( Fz )=O . /

PC Both Work ) = PCW ,nwc )

= PCW , )P( wz )

=(0.9 )( 0.9 )work

←

Atleast

onealarm

= 0.81 a\

1

PCW ,AFZ ) + PCF , nwz ) + Plwinwz )

,

PCW , )P( Fz ) + PC F.) PCWDTPCW . )P( Wz ) i

I ( 0.9 )( 0.1 ) + ( 0 . 1) ( 0.9 ) + 4.9) ( 0.9 ) I

.

- - . - . - .

. .

= 0.99 i

/ OR PC At least one ) = l - PC none )

= 1- ( 0 . 1) ( 0.1 )/

1-- - = 0.99 1

P ( C)= 0.7 PCN )= 0.3

PC At least one ) = 1 - PC none )

= 1 - P ( N,

nNz AN }nNynNsnN6 )

=L -PCN , )P( Ne ) PCN })P( Ny ) PCNSTPCNG )

=p - @.3) ( 0 . 3) ( 0 . 3) ( o . 3) CO . 3)

(0-3)=1- 0.36

= 1 - 7.29 E . 4←

1 - 0.000729

= 0.999271

J.M. Villalobos

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Conditional Probability

The probability that event A will occur given that event B already

happened is

P (A|B) =P (A \ B)

P (B)

Ex: Given P (L) = 0.5, P (H) = 0.4, P (L\H) = 0.3 find

10

:nil

puta) PCLIH ) =P # L

HyPCH )

.= 01=0.75 g. z

0.3 Oil

0.4

s

b) PIHK )=Pfy÷nY \o@z= 0-3 PCLNTT ) p( TH )

0.7=0.6

c) PC [ Itt ) =P(Ln# =

0-2=0.5Pct ) 0.6

J.M. Villalobos

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� 2016

Ex: The following table summarizes the car make of 300 ECC-CC

students.

- American Japanese German

Male 20 100 30Female 15 95 40

(a) Are the events: American Car and Male Independent events?

(b) Given that a randomly chosen car is German what is the prob-

ability that is own by a woman?

(c) Given that a randomly chosen student is Male what is the prob-

ability that he owns Japanese Car?

11

÷ °

-

3 5 l 9 5 70 / 300

?P ( An M ) = PCA ) P C M )

Soto±

335¥tat = BE ⇒ not In's 's.

petalsmyth = 4Is÷t= E

p 's in =p'fimm÷=i÷YET÷o

Pcm ' ⇒ = PYIIT = Taste :yes

J.M. Villalobos

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� 2016

Bayes Rule

Ex: Suppose that 46% of the population is male and that 82% own

a computer. If 71% of females own computers, what is the probability

that

(a) A randomly selected person does NOT own a computer?

(b) A randomly selected person owns a computer?

(c) If the person selected owns a computer, what is the probability

that the person was female?

12

0.82 C

§ 4.5 / Nt

µM ¥ p ( c ) f) = 0.71

\ ¥ c

0.54 F - NC0.2g

PCNC ) ==( 0.46)( 0.18 ) + ( 0.54 ) ( 0.29 )

Ee0.24

P ( C ) = 1 - PCNC )

= 1 - 0 . 24

= 0.76

Baye's

Rule

£

PCFIC ) = PciPCC )

( 0.54 ) ( 0.71 )= = = 0.5

0 . 76#

J.M. Villalobos

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� 2016

Ex: Suppose that 3% of the population has a rare disease. The

CDC has a test that is 98% e↵ective if the person has the disease and

97% e↵ective if the person does NOT have the disease. If a person is

a given this test what is the probability that

(a) the test will be positive?

(b) the test will be negative?

(c) If the test is positive, what is the probability that the person

actually has the the rare disease?

13

%'t '

D - t )

PC +7=0.03 ) (0-98)+10.97 ) ( 0.03 ) ¥0.02=0.0585\ #(+7=0.060.97

' H xqz f)

PH=1⇒-

1-0.06=0.94

P(D/t ) = PCD at ) ( 0.03 )( 0.98 )- = - = 0.5

pct ) 0.06

J.M. Villalobos

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� 2016

Counting

Fundamental Counting Principle

If event A can occur in m di↵erent ways and event B can occur in n

di↵erent ways then both events can occur in mn di↵erent ways.

Ex: SSN X X X X X X X X X

Ex: Phone Numbers (562) X X X X X X X

Ex: Lottery X X X X X X

14

§ 4.4 3 Kids : 2×2×2 = 8

⇐ 6×6=36

10 . 10 . 10 . 1 0 . 10 . 10.10.10 - 10 = (

09=1Billion

'

s

Obggmf →

334151049.10. LO . LO . ( o . 10 . 10 = 9

million

1 - 46 1 - 27

( Lotto ) ←

46.45 . 44.43-42 27 ==4Billi=Race : x × × x ×

5.4€ = 120 20 ! = 20.19.18 . . . .

. 2.1

5 !

J.M. Villalobos

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� 2016

Permutations

Combinations

Ex: This semester a math 150 class has 2 Republicans, 6 Democrats,

4 Independents, and 3 Communists. A 4-member committe will be

formed. What is the probability that:

(a) 1 Republican, 1 Democrat, and 2 Independent are chosen?

(b) 1 Republican and 3 Communist are chosen?

15

( order matters )

n Pri ,nn÷,,

1013=1,9÷= 109.8-7.6-5.4-3=7.6*3.2-1

= 720

( Order does not matter )

ncr = n!#n -51

.

loc }=

¥

3¥10 .9.8-7.6-5.4*1÷D7.6.5-4.3Is

=

12:15

PHR ,'D .2I)=2

.PH#2I0nC@PllR,3c) =

¥31915 ( 4

3-

J.M. Villalobos

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� 2016

16

6 C 3.9 (1

× × × × X

P( 3D ) =36

15 ( 4

C CH

L LL

A

PCTZIF , )=0

Pltz IT ,) > 0