Answer:
Option 1.
[tex]P(A) = \frac{1}{8},\ P(A|B) = \frac{1}{3}[/tex] Dependent
Option 2
[tex]P(A) = \frac{1}{4},\ P(A|B) = \frac{1}{4}[/tex] Independent
Option 3
[tex]P(B) = \frac{1}{8},\ P(B|A) = \frac{1}{4}[/tex] Dependent
Option 4
[tex]P(B) = \frac{1}{4},\ P(B|A) = \frac{1}{4}[/tex] Independent
Step-by-step explanation:
Two events A and B are independent if the occurrence of A does not affect the probability of B.
On the other hand The probability of A given B is defined as:
[tex]P (A | B) = \frac{P (A\ and\ B)}{P (B)}[/tex]
When two events are independent then:
[tex]P (A\ and\ B) = P (A) * P (B)[/tex]
So if the two events A and B are independent this means that:
[tex]P (A | B) = \frac{P (A) * P (B)}{P (B)}[/tex]
[tex]P (A | B) = P (A)[/tex]
Which makes sense because if the events are independent then the probability of A not being affected by B.
So to solve this problem identify in what cases
[tex]P (A | B) = P (A)[/tex] or [tex]P (B | A) = P (B)[/tex]
When this happens those events are independent
Option 1.
[tex]P(A) = \frac{1}{8},\ P(A|B) = \frac{1}{3}[/tex] Dependent
Option 2
[tex]P(A) = \frac{1}{4},\ P(A|B) = \frac{1}{4}[/tex] Independent
Option 3
[tex]P(B) = \frac{1}{8},\ P(B|A) = \frac{1}{4}[/tex] Dependent
Option 4
[tex]P(B) = \frac{1}{4},\ P(B|A) = \frac{1}{4}[/tex] Independent
