On February $28,2013$, a Field Poll Survey reported that $61\%$of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among $18\text{to}39$year olds (California registered voters), the approval rating was $78\%$. 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 those CA registered voters who support same-sex marriage, $75\%$say the ruling is important to them.

In this problem, let: •

C = California registered voters who support same-sex marriage. • B = 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 • A = California registered voters who are $18\text{to}39$years old.

a. Find P(C).

b. Find P(B).

c. Find P(C|A).

d. Find P(B|C).

e. In words, what is C|A?

f. In words, what is B|C?

g. Find P(C AND B).

h. In words, what is C AND B?

i. Find P(C OR B).

j. Are C and B mutually exclusive events? Show why or why not

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

Let the probability of California registered voters who support same-sex marriage is,

$$\begin{gathered} P\left(C\right)=\frac{61}{100} \\ P\left(C\right)=0.61 \end{gathered}$$

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

Let the probability of California registered voters who say the supreme court's ruling is,

$$\begin{gathered} P\left(B\right)=\frac{6}{10} \\ P\left(B\right)=0.6 \end{gathered}$$

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

Let the probability of California registered voters who are $18$to $39$years old is,

$$\begin{gathered} P\left(C|A\right)=\frac{78}{100} \\ P\left(C|A\right)=0.78 \end{gathered}$$

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

Let the probability of California registered voters who say the ruling is more important to them is,

$$\begin{gathered} P\left(B|C\right)=\frac{75}{100} \\ P\left(B|C\right)=0.75 \end{gathered}$$

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

The C|Ais California registered voters who support same-sex marriage and California registered voters who are$18\text{to}39$years old.

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

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

California registered voters who are $18\text{to}39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

The B|C is California registered voters who say the supreme court ruling and California registered voters who support same-sex marriage.

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

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

California registered voters who are $18$to $39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

Let the probability of P (C AND B) is,

$$\begin{gathered} P\left(CANDB\right)=P\left(B|C\right)P\left(C\right) \\ P\left(CANDB\right)=\left(0.75\right)\left(0.61\right) \\ P\left(CANDB\right)=0.46 \end{gathered}$$

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

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

California registered voters who are $18$to $39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

The C and B in words are California registered voters who support same-sex marriage and California registered voters who say the ruling is important to them.

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

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

California registered voters who are $18$to $39$years old is $78\%$

California registered voters who say the ruling is important to them is $75\%$

Let the probability of P (C or B) is,

$$\begin{gathered} P\left(CORB\right)=P\left(B\right)+P\left(C\right)-P\left(CANDB\right) \\ P\left(CORB\right)=\left(0.6\right)+\left(0.61\right)-0.46 \\ P\left(CORB\right)=0.75 \end{gathered}$$

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

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

California registered voters who are $18$to $39$years old is $78\%$

California registered voters who say the ruling is important to them is$75\%$

To show Cand B are mutually exclusive, there should not be any common events in between them (that is) $P\left(CANDB\right)=0$

Here,$P\left(CANDB\right)=0.46$