Question

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

probabilities	independent	dependent
(b) $p(a)=\frac{1}{3}$; $p(b)=\frac{1}{8}$; $p(a	b)=\frac{1}{3}$	$\circ$	$\circ$
(c) $p(a)=\frac{1}{7}$; $p(b)=\frac{1}{2}$; $p(a	b)=\frac{1}{4}$	$\circ$	$\circ$
(d) $p(a)=\frac{1}{5}$; $p(b)=\frac{1}{7}$; $p(b	a)=\frac{1}{3}$	$\circ$	$\circ$

Explanation:

To determine if events \( A \) and \( B \) are independent, we use the definition of independence: \( P(A \text{ and } B) = P(A) \times P(B) \) (for part a) or \( P(A|B) = P(A) \) (for parts b, c, d) or \( P(B|A) = P(B) \) (for part d).

Part (a)

Step 1: Calculate \( P(A) \times P(B) \)

\( P(A) = \frac{1}{3} \), \( P(B) = \frac{1}{2} \), so \( P(A) \times P(B) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \).

Step 2: Compare with \( P(A \text{ and } B) \)

Given \( P(A \text{ and } B) = \frac{1}{6} \), which equals \( P(A) \times P(B) \). So independent.

Part (b)

Step 1: Check \( P(A|B) \) vs \( P(A) \)

Given \( P(A|B) = \frac{1}{3} \) and \( P(A) = \frac{1}{3} \).

Step 2: Determine independence

Since \( P(A|B) = P(A) \), events are independent.

Part (c)

Step 1: Check \( P(A|B) \) vs \( P(A) \)

Given \( P(A|B) = \frac{1}{4} \) and \( P(A) = \frac{1}{7} \).

Step 2: Determine independence

\( \frac{1}{4}
eq \frac{1}{7} \), so dependent.

Part (d)

Step 1: Check \( P(B|A) \) vs \( P(B) \)

Given \( P(B|A) = \frac{1}{3} \) and \( P(B) = \frac{1}{7} \).

Step 2: Determine independence

\( \frac{1}{3}
eq \frac{1}{7} \), so dependent.

Answer:

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