Question

On February $28$, 2013, a Field Poll Survey reported thatlocalid="1649368651174" $61\%$of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Amonglocalid="1649369008817" $18$tolocalid="1649369026715" $39$year olds (California registered voters), the approval rating waslocalid="1649368659520" $78\%$. Six in ten California registered voters said that the upcoming Supreme Court's ruling about the constitutionality of California's Proposition $8$was either very or somewhat important to them. Out of thoselocalid="1649369095342" $\mathrm{CA}$registered voters who support same-sex marriage,localid="1649368674680" $75\%$say the ruling is important to them.

In this problem, let:

- localid="1649368705592" $\mathrm{C}$localid="1649368709078" $=$California registered voters who support same-sex marriage.

- $\mathrm{B}$localid="1649368720636" $=$California registered voters who say the Supreme Court's ruling about the constitutionality of California's Proposition $8$is very or somewhat important to them

- $\mathrm{A}$$=$California registered voters who are$18$to$39$years old.

a. Find $\mathrm{P}\left(\mathrm{C}\right)$.

b. Find $\mathrm{P}\left(\mathrm{B}\right)$.

c. Find $\mathrm{P}(\mathrm{C}\mid \mathrm{A})$.

d. Find$\mathrm{P}(\mathrm{B}\mid \mathrm{C})$.

e. In words, what is$\mathrm{C}\mid \mathrm{A}$localid="1649368754274" $?$

f. In words, what is$\mathrm{B}\mid \mathrm{C}$localid="1649368797480" $?$

g. Findlocalid="1649368804533" $\mathrm{P}(\mathrm{C}\mathrm{AND}\mathrm{B})$.

h. In words, what islocalid="1649368127412" $\mathrm{C}$localid="1649368441214" $\mathrm{AND}$localid="1649368191475" $\mathrm{B}$$?$

i. Findlocalid="1649368117801" $\mathrm{P}(\mathrm{C}\mathrm{OR}\mathrm{B})$.

j. Arelocalid="1649368817644" $\mathrm{C}$andlocalid="1649368811844" $\mathrm{B}$mutually exclusive eventslocalid="1649368076130" $?$Show why or why not.

Short answer

a.

b.

c.

d.

e,

f.

g.

h.

i.

j.

Step-by-step solution

Finding of P(C) (part a)

a.

California registered voters who support same-sex marriage is $61\%$.

California registered voters who say the ruling is important to them is

The likelihood of event$\mathrm{C}$is to find the probability that a California registered voter supports same-sex weddings.

So,

$\mathrm{P}\left(\mathrm{C}\right)=\frac{61}{10}$

$\mathrm{P}\left(\mathrm{C}\right)=0.61$

Finding of P(B) (part b)

b.

California registered voters who say the supreme court ruling is$6\%$

The event $\mathrm{B}$occurs when a California registered voter thinks the Supreme Court's decision on the validity of Proposition $8$in California is very or somewhat important to them.

As a result, the likelihood of event$\mathrm{B}$is:

$\mathrm{P}\left(\mathrm{B}\right)=\frac{6}{10}$

$\mathrm{P}\left(\mathrm{B}\right)=0.6$

Finding of P(C∣A) (part c)

c.

California registered voters who are years old is$78\%$

We must calculate the probability of event $\mathrm{C}$given $\mathrm{A}$, where $\mathrm{A}$is the probability of a voter being between the ages of $18$and $39$:

$\mathrm{P}(\mathrm{C}\mid \mathrm{A})=\frac{78}{100}$

$\mathrm{P}(\mathrm{C}\mid \mathrm{A})=0.78$