lec 18 probability putting rules to work

7/24/2019 Lec 18 Probability Putting Rules to Work

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Math 361

Probability & Statistic


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Example Problems


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Whats the chance of hitting a red light at least once during the days)? (assuming P(R) = 0.5! P(") = 0.0#! P($) = 0.%&)

'irst roach

We could start by calculating the probability of hitting exactly 1 red lighweek, then exactly 2, then 3, 4, and 5

Then wed !ust need to add the" up#

$%1 &ed' ( 5 x %&1x )&4' ( 5 x %*+351x *+54' ( 0.&*#

$%2 &ed' ( 1* x %&2x )&3' ( 1* x %*+352x *+53' ( 0.%#

$%3 &ed' ( 1* x %&3x )&2' ( 1* x %*+353x *+52' ( 0.&+&&

$%4 &ed' ( 5 x %&4x )&1' ( 5 x %*+354x *+51' ( 0.0#++

$%5 &ed' ( *+35 x *+35 x *+35 x *+35 x *+35 ( 0.005*

P(t least & Red) = 0.&*# , 0.%# , 0.&+&& , 0.0#++ , 0.005* = 0.


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-econd roach

-t can be easier to work using the co"ple"ent of the e.ent were interested i

Whats the co"ple"ent of getting at least one red light/

-ts getting no red lights, fi.e days in a row+

We know the probability of hitting a red light is *+35 each day, so by the co"

the probability of not hitting a red light each day is & 0.5 = 0.%5

The probability of "aking it through fi.e days in a row without hitting a red lig

)ow, taking the co"ple"ent of this co"pound e.ent, we find the probability

happen% no * red lights' is0

P(at least & red light) = & / (0.%5)5= & 0.&&% = 0.++#


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Whats the roaility 1 dont encounter a red light until Wed(assuming P(R) = 0.5! P(") = 0.0#! P($) = 0.%&)

or that to happen, youd ha.e to see green or yellow on onday,

yellow on Tuesday, and then red on Wednesday

o"bining all those probabilities could get "essy+ We can si"plify

thinking of it as not red on onday and Tuesday and then red on W

$%&' ( 1 $%&' ( 1 *+35 ( *+5, so

$%& on on 6)7 & on Tue 6)7 & on Wed' ( $%&' x $%&' x $

( *+5 x *+5 x *+35 ( 0.+

3heres aout a .+4 chance that this week 1ll hit my first red

Wednesday morning


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-n 2**1, asterfoods, the "anufacturers of 8s9 "ilk chococandies, decided to add another color to the standard color lineyellow, red, orange, blue, and green+ To decide which color to asur.eyed kids in nearly e.ery country of the world and asked the

a"ong purple, pink, and teal+ The global winner was purple# -n t;tates, #*4of those who .oted said urle, 24said teal, andsaid ink+ et s use =apans percentages to ask so"e ?

1+ What s the probability that a =apanese 8s sur.ey respond

at rando" preferred either pink or teal/

2+ -f we pick two respondents at rando", what @s the probability t

both selected purple/

3+ -f we pick three respondents at rando", whats the probability

one preferred purple/


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The probabilities of an e.ent is its longAter" relati.e fre?uBere we are told the relati.e fre?uencies of the three resp

ake sure the probabilities are legiti"ate+

Bere, theyre not+ Cither there was a "istake, or the othe"ust ha.e chosen a color other than the three gi.en

$%pink' ( *+3D, $%teal' ( *+3, $%purple' ( *+1

Cach is between * and 1, but they dont all add up to 1

The re"aining 1*E of the .oters "ust ha.e not expressepreference or written in another color

Well put the" under Fno preferenceG, so $%no pref+' ( *+1

With this addition, we ha.e a legiti"ate assign"ent of pro


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6uestion &7 What s the roaility that a 8aanese 9sur;ey resondent selected at random referred eitheteal?

Plan7


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6uestion *. 1f we ick two resondents at random! whats the rothey oth said urle?

Plan7 3he word @othA suggests we want P( and B)! which calls9ultilication Rule. 3hink aout the assumtion.

1ndeendence ssumtion7 -t s unlikely that the choice "ade by one affected the choice of the other, so the e.ents see" to be independent+

ultiplication &ule+

9echanics7 -how your work

$%both purple'

( $%first respondent picks purple and second respondent picks purple'

( $%first respondent picks purple' x $%second respondent picks purple'

( *+1 x *+1 ( *+*25

>onclusion7 3he roaility that oth resondents ick urle is


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6uestion . -f we pick three respondents at rando", whaprobability that at least one preferred purple/

Plan7 3he hrase @at leastCA often flags a uestion

answered y looking at the comlement! and thats tharoach here.

The co"ple"ent of F6t least one preferred purple is Fnone

preferred purpleG

$%at least on picked purple' ( $%Jnone picked purpleKc'

( 1 $%none picked purple'

$%none picked purple' ( $%not purple and not purple and n

These are independent e.ents because they are choicesrando" respondents so we can use the "ultiplication rule


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9echanics7

$%none picked purple' ( $%first not purple' x $%second no$%third not purple'

( L$%not purple'M3+ $%not purple' ( 1 A $%purple' ( 1 *+1 ( *+D4

;o $%none picked purple' ( %*+D4'3( *+5N2I+

$%at least 1 picked purple'

( 1 A $%none picked purple'

( 1 A *+5N2I ( *+4*I3+

>onclusion7 3heres aout a #0.24 chance that at least

the resondents icked urle


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6. 3he 9asterfoods comany says that efore the introduction o

yellow candies made u *04 of their lain 9:9 s! red another *0

orange! lue! and green each made u &04. 3he rest were rown

a' -f you pick an 8 at rando", what is the probability that

it is brown/

*+3*

it is yellow or orange/

*+3* it is not green/

*+N*

it is striped/

*+*


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b' -f you pick three 8 s in a row, what is the probability

they are all brown/

*+*2I

the third one is the first one that s red/

*+12D

none are yellow/

*+512

at least one is green/

*+2I1


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6. "ou roll a fair die three times. What is the roaility

Oou roll all s/

*+**4

Oou roll all odd nu"bers/

*+125

)one of your rolls gets a nu"ber di.isible by 3/

*+2N

Oou roll at least one 5/

*+421

The nu"bers you roll are not all 5s/

*+NN5


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The End

lec 18 probability putting rules to work

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