Answer: Option D
[tex]P (C) =P (C | A) = 0.55[/tex]
Step-by-step explanation:
First we assign names to events:
Event C: The selected people are from California
Event A: The selected person prefers the A mark
Now notice in the table that the total number of people is: 275.
Then, the number of people who prefer the A mark is: 176
The number of people who are from California is: 150
The number of people in California who prefer the A brand is: 96
Then we have that:
[tex]P(C)=\frac{150}{275}=\frac{6}{11}=0.55[/tex]
[tex]P(A)=\frac{176}{275}=\frac{16}{25}[/tex]
[tex]P (C\ and\ A) = \frac{96}{275}=0.3491[/tex]
Then:
[tex]P(C|A)=\frac{P(C\ and\ A)}{P(A)}\\\\P(C|A)=\frac{\frac{96}{275}}{\frac{16}{25}}=\frac{6}{11}=0.55[/tex]
By definition two events C and A are independent if and only if:
[tex]P (C\ and\ A) = P (A)*P (C)[/tex]
Then, if A and C are independent events, it must be fulfilled that:
[tex]P(C|A)=\frac{P(A)*P(C)}{P(A)}\\\\P(C|A)=P(C)[/tex]
Note that [tex]P (C) = 0.55[/tex] and [tex]P (C | A) = 0.55[/tex]
So:
[tex]P (C | A) = P (C)[/tex]
Therefore the events are independent
The answer is option D
