Binomial and Geometric Probability Key

8/20/2019 Binomial and Geometric Probability Key

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Binomial and Geometric Probability Practice Problems KEY

Directions: Use proper probability notation to transfer from the verbal question. Write the formula with

values plued in! then select the appropriate distribution function on the calculator and enter appropriatevalues" Write the numerical answer followin the formula #D$%&' write calculator spea()*"

+" $n the ,-'! there are five answer choices #-! B! .! D! and E*" 'he probability of randomly uessin

the correct answer is "/"

a* What is the probability that on a /012uestion section of the ,-' by complete random uessinthat e3actly 4 2uestions will be answered correctly5

Binomial /0=n /"= p 67/8"*4#"*/#"4

/0*4#

+94 =

== X P

b* What is the probability that on a /012uestion section of the ,-' by complete random uessin

that 7 or fewer 2uestions will be answered correctly5

Binomial /0=n /"= p 9466"*4#"*/#"7

/0*4#"*/#"

6

/0*7#

+:7/06 =

+⋅⋅⋅+

=≤ X P

c* What is the probability that on a /012uestion section of the ,-' by complete random uessin

that the first correctly uessed answered is the fourth5

Geometric /"= p +6/;"*/#"*4#"*;# 8 === X P

d* What is the probability that on a /012uestion section of the ,-' by complete random uessinthat the first correct answer will be within the first 7 uesses5

Geometric /"= p 989:"*/#"*4#"*/*#"4#"/"*7# 0 =+⋅⋅⋅++=≤ X P

e* What is the e3pected number of correct uesses on a /012uestion section of the ,-' e3am5

Binomial /0=n /"= p 0*/*#"/0# === np µ 2uestions

f* What is the standard deviation of correct uesses on a /012uestion section of the ,-' e3am5

Binomial /0=n /"= p /*4*#"/*#"/0#*+# ==−= pnpσ 2uestions



/" <a=or universities claim that 9/> of their senior athletes raduate that year" ?ifty senior athletic students

attendin ma=or universities are randomly selected and recorded in order of selection"a* What is the probability that e3actly ;6 senior athletic students raduate that year5

Binomial 06=n 9/"= p 60:4"*/4#"*9/#";6

06*;6#

+6;6 =

== X P

b* What is the probability that ;6 or ;+ or ;/ senior athletic students raduated that year5

Binomial 06=n 9/"= p

++9:"*/4#"*9/#";/06*/4#"*9/#"

;+06*/4#"*9/#"

;606*;/;6#

4;/:;++6;6 =

+

+

=≤≤ X P

c* What is the probability that ;6 or fewer senior athletic students raduated that year5

Binomial 06=n 9/"= p :/76"*/4#"*9/#";6

06*/4#"*9/#"

6

06*;6#

+6;6066 =

+⋅⋅⋅+

=≤ X P

d* What is the probability that ;+ or more senior athletic students raduated that year5

Binomial 06=n 9/"= p;+ 06 6

06 06# ;+* #"9/* #"/4* #"9/* #"/4* "69;6

;+ 06 P X

≥ = + ××× + = ÷ ÷

e* What is the probability that ;6 or more senior athletic students raduated that year5

Binomial 06=n 9/"= p;6 +6 06 6

06 06# ;6* #"9/* #"/4* #"9/* #"/4* "+889

;6 06 P X

≥ = + ×××+ = ÷ ÷

f* What is the probability that the first senior athletic student to raduate in the roup of 06 that

year is the 0th selected5

Geometric 9/"= p 66;;"*9/#"*/4#"*0# ; === X P

* What is the probability that the first senior athletic student to raduate in the roup of 06 thatyear is the 86th selected5

Geometric 9/"= p +9/:+674/"7*9/#"*/4#"*86#

−×=== X P

h* What is the probability that the first senior athletic student to raduate in the roup of 06 thatyear is within the first +6 selected5

Geometric 9/"= p :::::9"*9/#"*/4#"*9/*#"/4#"9/"*+6# : =+⋅⋅⋅++=≤ X P

i* What is the e3pected number of senior athletic students to raduate that year5

Binomial 06=n 9/"= p 87*9/*#"06# === np µ students

=* What is the standard deviation of senior athletic students raduatin that year5

Binomial 06=n 9/"= p +90"8*/4*#"9/*#"06#*+# ==−= pnpσ students



8" Will ?umble is the only receiver for his @, football team with the li(elihood of catchin a pass of "+0"

a* What is the probability that / passes are cauht out of 7 passes5

Binomial 7=n +0"= p +97/"*40#"*+0#"/

7*/#

;/ =

== X P

b* What is the probability that no passes are cauht out of 7 passes5

Binomial 7=n +0"= p 899+"*40#"*+0#"6

7*6#

76 =

== X P

c* What is the probability that only 6 or + pass is cauht out of 7 passes5

Binomial 7=n +0"= p 9970"*40#"*+0#"+

7*40#"*+0#"

6

7*+#*6#

0+76 =

+

==+= X P X P

d* What is the probability that / or fewer passes are cauht out of 7 passes5

Binomial 7=n +0"= p :0/9"*40#"*+0#"/

7*40#"*+0#"

+

7*40#"*+0#"

6

7*/#

;/0+76 =

+

+

=≤ X P

e* What is the probability that more than / passes are cauht out of 7 passes5

Binomial 7=n +0"= p 8 8 7 67 7# /* #"+0* #"40* #"+0* #"40* "6;988 7

P X > = + + = ÷ ÷

L

f* What is the probability that the first pass cauht is on the +st pass5

Geometric +0"= p +0"*+# == X P

* What is the probability that the first pass cauht is on the ;th pass5

Geometric +0"= p 6:/+"*+0#"*40#"*;# 8 === X P

h* What is the probability that the first pass is cauht within the first 8 attempts5

Geometric +0"= p 840:"*+0#"*40#"*+0*#"40#"+0"*8# / =++=≤ X P

i* What is the probability that the first pass is cauht after the first 8 attempts5

Geometric +0"= p 7+;+"*+0#"*40#"*+0*#"40#"+0"+*8#+*8# /

=++−=≤−=> X P X P

=* What is the e3pected number of catches with 7 attempts5

Binomial 06=

n 9/"= p :"*+0*#"7# === np µ catches

(* What is the e3pected number of attempts for the first pass cauht5

Geometric +0"= p 79"7+0"

++===

p µ attempts



;" -ccordin to a recent .ensus Bureau report! +/"9> of -mericans live below the poverty level" ,uppose you

plan to sample at random +66 -mericans and count the number of people who live below the poverty level"What is the probability that you count +6 or fewer5

Binomial +66n = "+/9 p = 6 +66 +6 6

+66 +66# +6* #"+/9* #"498* #"+/9* #"498* "/7+;

6 +6 P X

≤ = + ×××+ = ÷ ÷

0" Af you roll a pair of dice! what are the chances of ettin doubles for the first time on the 0th roll5

Geometric7 +

87 7 p = =

;0 +

# 0* "646;7 7 P X = = = ÷ ÷

7" - potential buyer will sample videotapes from a lare lot of new videotapes" Af she finds at least onedefective one! she&ll re=ect the entire lot" Af ten percent of the lot is defective! what is the probability that she&ll

find a defective tape by the ;th videotape5

Geometric "+ p = / 8# ;* "+ #"*#"+* #"* #"+* #"* #"+* "8;8 P X ≤ = + + + =

9" $ne in +6 hih school raduates in the state of ?lorida sends an application to the University of ?lorida!Csays the direct of admissions" Af ?lorida has +/6!666 hih school raduates ne3t year! how many applicants

would they e3pect5

Binomial +/6666n = "+ p = #+/6666*#"+* +/666np µ = = = applicants

4" - ame is played with a spinner on a circle! li(e the minute hand on a cloc(" 'he circle is mar(ed evenly

from + to +66" 'he player spins the spinner and the resultin number is the number of seconds the player isiven to solve a randomly selected mathematics problem" ,uppose there are 86 students playin in the class"

What is the probability that +6 of the students received over a minute to solve the problem5

Binomial 86n = "; p =+6 /686

# +6* #";* #"7* "++0/+6

P X = = = ÷

What is the probability that half of the students received 86 or less seconds5

Binomial 86n = "8 p = +0 +086

# +0* #"8* #"9* "6+67+0

P X

= = = ÷

What is the probability that the fourth student was the first to receive a minute or less5

Geometric "7 p =8# ;* #";* #"7* "684; P X = = =

What is the probability that at least half of the students received ;0 seconds or more5

Binomial 86n = "07 p = +0 86 86 686 86

# +0* #"07* #";;* #"07* #";;* "46+7+0 86

P X

≥ = + + = ÷ ÷

L

What is the probability that the first student to receive a minute or more was the +6th student5

Geometric ";+ p = ;# +6* #";+* #"0* "+"8/ +6 P X −= = = ×



" - wildlife bioloist e3amines duc(s for a enetic trait he suspects may be lin(ed to sensitivity to industrial

to3ins in the environment" Previous research had established that this trait is found in one of every 4 duc(s" @ecollects and e3amines +/ duc(s" -ssume the fre2uency of the trait has not chaned"

a* What is the probability he finds the trait in none of the duc(s5

+/# 6* #"490* "/6+; P X = = =

b* What is the probability he finds the trait in at least one of the duc(s5

+/#at least one* + #none* + #"490* "947 P P = − = − =

c* What is the probability that the first duc( he finds the trait in is the 9th duc( he e3amines5

Geometric "+/0 p = ( ) ( )7

# 9* "490 "+/0 "607+ P X = = =

d* What is the probability he finds the trait in e3actly 9 duc(s5

Binomial +/n = "+/0 p = 9 0 ;+/

# 9* #"+/0* #"490* +"89 +69

P X −

= = = × ÷

e* What is the probability he finds the trait in 8 or ; duc(s5

Binomial +/n = "+/0 p = 8 ; 4+/ +/

# 8 or ;* #"+/0* #"490* #"+/0* #"490* "+9698 ;

P X

= = + = ÷ ÷

f* What is the probability that the first duc( he finds the trait in is within the first 0 duc(s5

Geometric "+/0 p =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )/ 8 ;

# 0* "+/0 "490 "+/0 "490 "+/0 "490 "+/0 "490 "+/0 ";49+ P X ≤ = + + + + =

* What is the probability he finds the trait in no more than ; duc(s5

Binomial +/n = "+/0 p = 6 +/ ; 4+/ +/

# ;* #"+/0* #"490* #"+/0* #"490* "4496 ;

P X

≤ = + ××× + = ÷ ÷

h* What is the probability he finds the trait in at least 9 duc(s5

Binomial +/n = "+/0 p = 9 0 +/ 6 ;+/ +/

# 9* #"+/0* #"490* #"+/0* #"490* /"+/+ +69 +/

P X −

≥ = + ×××+ = × ÷ ÷

i* @ow many duc(s do you e3pect him to find the trait in5 With what standard deviation5

Binomial +/n = "+/0 p = #+/*#"+/0* +"0 duc(snp µ = = =

#+ * #+/*#"+/0*#"490* +"+;7 duc(snp pσ = − = =

=* @ow many duc(s do you e3pect him to e3amine in order to find one with the trait5 With what standard

deviation5

Geometric "+/0 p =+ +

4 duc(s"+/0 p

µ = = =/ /

+ + "+/09";48 duc(s

#"+/0*

p

pσ

− −= = =

Binomial and Geometric Probability Key

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