Question

On February , a Field Poll Survey reported that

of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among
year olds (California registered voters), the approval rating was
. Six in ten California registered voters said that the upcoming Supreme Court’s ruling about the constitutionality of California’s Proposition
was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage,
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
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

Answer 1

a. P(C) =
`\( (3)/(5) \)`

b. P(B) =
`\( (3)/(5) \)`

c. P(C|A) =
`\( (3)/(5) \)`

d. P(B|C) = 1

e. C|A is the probability that a California registered voter supports same-sex marriage given that they are 18 years old.

f. B|C is the probability that a California registered voter considers the Supreme Court's ruling on Proposition 8 important given that they support same-sex marriage.

g. P(C AND B) =
`\( (3)/(5) \)`

h. C AND B is the event that a California registered voter both supports same-sex marriage and considers the Supreme Court's ruling on Proposition 8 important.

i. P(C OR B) =
`\( (3)/(5) \)`

j. No, C and B are not mutually exclusive events because there are individuals who both support same-sex marriage and consider the Supreme Court's ruling on Proposition 8 important.

Step-by-step explanation:

In the given scenario, the probability of California registered voters supporting same-sex marriage (P(C)) and the probability of them considering the Supreme Court's ruling on Proposition 8 important (P(B)) are both
`\( (3)/(5) \)`. This indicates a strong correlation between these two events.

The conditional probability P(C|A), which represents the likelihood of a California registered voter supporting same-sex marriage given that they are
`\(18\)` years old, is also
`\( (3)/(5) \)`. This suggests that age does not significantly impact the stance on same-sex marriage among California registered voters.

P(B|C) is 1, indicating that if a voter supports same-sex marriage, they are certain to consider the Supreme Court's ruling on Proposition 8 important.

The events C|A and B|C express the conditional probabilities in the context of age and support for same-sex marriage, and the importance of the Supreme Court's ruling, respectively.

The joint probability P(C AND B) is
`\( (3)/(5) \)`, representing the likelihood of a California registered voter supporting same-sex marriage and considering the Supreme Court's ruling on Proposition 8 important simultaneously.

Finally, P(C OR B) is also
`\( (3)/(5) \)`, implying that a significant portion of California registered voters either supports same-sex marriage, considers the Supreme Court's ruling important, or both. Notably, C and B are not mutually exclusive events, as individuals may fall into both categories.