Question

In: Math

1. Suppose that Superman is the favorite hero of 3 5% of all DC Comics fans,...

1.

Suppose that

Superman is the favorite hero of

3

5%

of

all DC Comics fans, Batman is the

favorite of 26% of fans, Wonder Woman is the favorite of 19%, Green Lantern is the favorite

of 12%, and the

Flash is the favorite of all the rest of fans

.

a)

If a

DC Comics fan

is selected at random, what is the

probability that

:

i.

T

he Flash is that person’s favorite superhero

?

ii.

T

he person’s favorite superhero is Batman or Wonder Woman?

iii.

T

he person’s

favorite superhero is not Superman

?

b)

If you were to randomly select

five

DC Comics fans

, what is the probability that:

i.

Batman

is the favorite superhero of all five people

?

2

ii.

No

ne of the

five

people

identify Green Lantern as their favorite superhero

?

iii.

All five people identify

Green Lantern

as their favorite superhero

or

all five identify

Wonder Woman

as their favorite

?

iv.

N

ot

all

five

people identify

the Flash

as their favorite superhero

?

v.

Superman is the favorite superhero of

at least

one of the

five

people

?

Solutions

Expert Solution

It is given that Superman is the favorite hero of 35% of all DC Comics fans, Batman is the favorite of 26% of fans, Wonder Woman is the favorite of 19%, Green Lantern is the favorite of 12%, and the Flash is the favorite of all the rest of fans i.e. 100%-35%-26%-19%-12% = 8% of the fans.

If a DC Comics fan is selected at random, the probability that the Flash is that person’s favorite superhero = 8% = 0.08

If a DC Comics fan is selected at random, the probability that the person’s favorite superhero is Batman or Wonder Woman

= Probability that the person’s favorite superhero is Batman + Probability that the person’s favorite superhero is Wonder Woman

= 26% + 19% = 0.26 + 0.19 = 0.45

If a DC Comics fan is selected at random, the probability that the person’s favorite superhero is not Superman

= 1 - Probability that the person’s favorite superhero is Superman

= 1 - 35% = 1 - 0.35 = 0.65

.

  Assumption: All the 5 DC Comics fans were selected independently.

If 5 DC Comics fans were randomly selected, the probability that Batman is the favorite superhero of all the 5 people

= (Probability that Batman is the favorite superhero of 1 randomly selected DC fan)5  

= (26%)5 = (0.26)5 = 0.0011881376

If 5 DC Comics fans were randomly selected, the probability that none of the 5 people identify Green Lantern as their favorite superhero

= (Probability that 1 randomly selected DC Comics fan do not identify Green Lantern as the person's favorite superhero)5  

= (1 - Probability that 1 randomly selected DC Comics fan identify Green Lantern as the person's favorite superhero)5  

= (1 - 12%)5 = (1 - 0.12)5 = (0.88)5 = 0.5277319168

If 5 DC Comics fans were randomly selected, the probability that all of the 5 people identify Green Lantern as their favorite superhero or all the 5 people identify Wonder Woman as their favorite superhero

= (Probability that 1 randomly selected DC Comics fan identify Green Lantern as the person's favorite superhero)5 + (Probability that 1 randomly selected DC Comics fan identify Wonder Woman as the person's favorite superhero)5  

= (12%)5 + (19%)5 = (0.12)5 + (0.19)5 = 0.0002724931

If 5 DC Comics fans were randomly selected, the probability that not all of the 5 people identify the Flash as their favorite superhero

= Probability that at least 1 of the 5 randomly selected DC Comics fans do not identify the Flash as the person's favorite superhero

= 1 - Probability that all of the 5 DC Comics fans identify the Flash as their favorite superhero

= 1 - (Probability that 1 randomly selected DC Comics fan identify the Flash as the person's favorite superhero)5

= 1 - (8%)5 = 1 - (0.08)5 = 1 - 0.0000032768 = 0.9999967232

If 5 DC Comics fans were randomly selected, the probability that Superman is the favorite superhero of at least 1 of the 5 people

= 1 - Probability that Superman is the favorite superhero of none of the 5 randomly selected DC Comics fans

= 1 - (Probability that Superman is not the favorite superhero of 1 randomly selected DC Comics fan)5

= 1 - (1 - 35%)5 = 1 - (1 - 0.35)5 = 1 - (0.65)5 = 1 - 0.1160290625 = 0.8839709375

milcah answered 2 hours ago
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