Question

question
for each set of probabilities, determine whether the events ( a ) and ( b ) are independent or dependent.

probabilities	independent	dependent
(b) ( p(a)=\frac{1}{8}; p(b)=\frac{1}{4}; p(b	a)=\frac{1}{4} )	( circ )	( circ )
(c) ( p(a)=\frac{1}{8}; p(b)=\frac{1}{5}; p(a	b)=\frac{1}{3} )	( circ )	( circ )
(d) ( p(a)=\frac{1}{4}; p(b)=\frac{1}{3}; p(a \text{ and } b)=\frac{1}{12} )	( circ )	( circ )

Explanation:

Step1: Recall independence rule

Two events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A) \times P(B)$, or equivalently $P(A|B)=P(A)$ / $P(B|A)=P(B)$.

Step2: Solve part (a)

Check if $P(A|B)=P(A)$:
$P(A)=\frac{1}{4}$, $P(A|B)=\frac{1}{4}$. Since $P(A|B)=P(A)$, events are independent.

Step3: Solve part (b)

Check if $P(B|A)=P(B)$:
$P(B)=\frac{1}{4}$, $P(B|A)=\frac{1}{4}$. Since $P(B|A)=P(B)$, events are independent.

Step4: Solve part (c)

Check if $P(A|B)=P(A)$:
$P(A)=\frac{1}{8}$, $P(A|B)=\frac{1}{3}$. Since $\frac{1}{8}
eq \frac{1}{3}$, events are dependent.

Step5: Solve part (d)

Check if $P(A \text{ and } B)=P(A) \times P(B)$:
$P(A) \times P(B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$, which equals $P(A \text{ and } B)=\frac{1}{12}$. Events are independent.

Answer:

(a) Independent
(b) Independent
(c) Dependent
(d) Independent