probability basics probability experiment ... · §4.1 probability basics • probability...
TRANSCRIPT
J.M. Villalobos
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� 2016
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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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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� 2016
§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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� 2016
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
c
� 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
c
� 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