Question

for each set of probabilities, determine whether the events a and b are independent or dependent.
probabilities \tindependent \tdependent
(a) ( p(a) = \frac{1}{2} ); ( p(b) = \frac{1}{6} ); ( p(a|b) = \frac{1}{2} ) \t○ \t○
(b) ( p(a) = \frac{1}{6} ); ( p(b) = \frac{1}{4} ); ( p(a|b) = \frac{1}{5} ) \t○ \t○
(c) ( p(a) = \frac{1}{2} ); ( p(b) = \frac{1}{4} ); ( p(a \text{ and } b) = \frac{1}{8} ) \t○ \t○
(d) ( p(a) = \frac{1}{9} ); ( p(b) = \frac{1}{2} ); ( p(b|a) = \frac{1}{2} ) \t○ \t○

Explanation:

To determine if events \( A \) and \( B \) are independent, we use the multiplication rule for independent events:
Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
Alternatively, we can also check if \( P(A|B) = P(A) \) (or \( P(B|A) = P(B) \)), since conditional probability \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).

Part (a)

Given: \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{6} \), \( P(A|B) = \frac{1}{2} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(A|B) = P(A) \).

Step 2: Compare \( P(A|B) \) and \( P(A) \).

Here, \( P(A|B) = \frac{1}{2} \) and \( P(A) = \frac{1}{2} \).
Since \( P(A|B) = P(A) \), events \( A \) and \( B \) are independent.

Part (b)

Given: \( P(A) = \frac{1}{6} \), \( P(B) = \frac{1}{4} \), \( P(A|B) = \frac{1}{5} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(A|B) = P(A) \).

Step 2: Compare \( P(A|B) \) and \( P(A) \).

Here, \( P(A|B) = \frac{1}{5} \) and \( P(A) = \frac{1}{6} \).
Since \( \frac{1}{5}
eq \frac{1}{6} \), events \( A \) and \( B \) are dependent.

Part (c)

Given: \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{4} \), \( P(A \cap B) = \frac{1}{8} \)

Step 1: Calculate \( P(A) \cdot P(B) \).

\( P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \).

Step 2: Compare \( P(A \cap B) \) and \( P(A) \cdot P(B) \).

Here, \( P(A \cap B) = \frac{1}{8} \) and \( P(A) \cdot P(B) = \frac{1}{8} \).
Since \( P(A \cap B) = P(A) \cdot P(B) \), events \( A \) and \( B \) are independent.

Part (d)

Given: \( P(A) = \frac{1}{9} \), \( P(B) = \frac{1}{2} \), \( P(B|A) = \frac{1}{2} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(B|A) = P(B) \).

Step 2: Compare \( P(B|A) \) and \( P(B) \).

Here, \( P(B|A) = \frac{1}{2} \) and \( P(B) = \frac{1}{2} \).
Since \( P(B|A) = P(B) \), events \( A \) and \( B \) are independent.

Final Answers:

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

Answer:

To determine if events \( A \) and \( B \) are independent, we use the multiplication rule for independent events:
Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
Alternatively, we can also check if \( P(A|B) = P(A) \) (or \( P(B|A) = P(B) \)), since conditional probability \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).

Part (a)

Given: \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{6} \), \( P(A|B) = \frac{1}{2} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(A|B) = P(A) \).

Step 2: Compare \( P(A|B) \) and \( P(A) \).

Here, \( P(A|B) = \frac{1}{2} \) and \( P(A) = \frac{1}{2} \).
Since \( P(A|B) = P(A) \), events \( A \) and \( B \) are independent.

Part (b)

Given: \( P(A) = \frac{1}{6} \), \( P(B) = \frac{1}{4} \), \( P(A|B) = \frac{1}{5} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(A|B) = P(A) \).

Step 2: Compare \( P(A|B) \) and \( P(A) \).

Here, \( P(A|B) = \frac{1}{5} \) and \( P(A) = \frac{1}{6} \).
Since \( \frac{1}{5}
eq \frac{1}{6} \), events \( A \) and \( B \) are dependent.

Part (c)

Given: \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{4} \), \( P(A \cap B) = \frac{1}{8} \)

Step 1: Calculate \( P(A) \cdot P(B) \).

\( P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \).

Step 2: Compare \( P(A \cap B) \) and \( P(A) \cdot P(B) \).

Here, \( P(A \cap B) = \frac{1}{8} \) and \( P(A) \cdot P(B) = \frac{1}{8} \).
Since \( P(A \cap B) = P(A) \cdot P(B) \), events \( A \) and \( B \) are independent.

Part (d)

Given: \( P(A) = \frac{1}{9} \), \( P(B) = \frac{1}{2} \), \( P(B|A) = \frac{1}{2} \)

Step 1: Recall the definition of independence.

Events are independent if \( P(B|A) = P(B) \).

Step 2: Compare \( P(B|A) \) and \( P(B) \).

Here, \( P(B|A) = \frac{1}{2} \) and \( P(B) = \frac{1}{2} \).
Since \( P(B|A) = P(B) \), events \( A \) and \( B \) are independent.

Final Answers:

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