To determine whether events $A$ and $B$ are independent or dependent, we use the definition of independence:

$P(A \cap B) = P(A) \cdot P(B)$

If this equality holds, then $A$ and $B$ are independent; otherwise, they are dependent.

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(a) $P(A) = \frac{1}{6}, P(B) = \frac{1}{5}, P(B|A) = \frac{1}{5}$

Using the conditional probability formula:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

$\frac{1}{5} = \frac{P(A \cap B)}{1/6}$

Solving for $P(A \cap B)$:

$P(A \cap B) = \frac{1}{5} \times \frac{1}{6} = \frac{1}{30}$

Now, checking independence:

$P(A) \cdot P(B) = \frac{1}{6} \times \frac{1}{5} = \frac{1}{30}$

Since $P(A \cap B) = P(A) \cdot P(B)$, $A$ and $B$ are independent.

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(b) $P(A) = \frac{1}{5}, P(B) = \frac{1}{2}, P(A|B) = \frac{1}{10}$

Using:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

$\frac{1}{10} = \frac{P(A \cap B)}{1/2}$

Solving for $P(A \cap B)$:

$P(A \cap B) = \frac{1}{10} \times \frac{1}{2} = \frac{1}{20}$

Checking independence:

$P(A) \cdot P(B) = \frac{1}{5} \times \frac{1}{2} = \frac{1}{10}$

Since $P(A \cap B) \neq P(A) \cdot P(B)$ (because $\frac{1}{20} \neq \frac{1}{10}$), $A$ and $B$ are dependent.

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(c) $P(A) = \frac{1}{5}, P(B) = \frac{1}{3}, P(A \cap B) = \frac{1}{2}$

Checking independence:

$P(A) \cdot P(B) = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15}$

Since $P(A \cap B) \neq P(A) \cdot P(B)$ (because $\frac{1}{2} \neq \frac{1}{15}$), $A$ and $B$ are dependent.

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(d) $P(A) = \frac{1}{4}, P(B) = \frac{1}{9}, P(A \cap B) = \frac{1}{5}$

Checking independence:

$P(A) \cdot P(B) = \frac{1}{4} \times \frac{1}{9} = \frac{1}{36}$

Since $P(A \cap B) \neq P(A) \cdot P(B)$ (because $\frac{1}{5} \neq \frac{1}{36}$), $A$ and $B$ are dependent.

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Final Answers:

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

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Would you like a step-by-step breakdown for any of these?

Related Questions:

1. How do you calculate conditional probability using $P(A|B)$?
2. What is the difference between independent and mutually exclusive events?
3. How does Bayes' theorem relate to conditional probability?
4. Can two events be both independent and mutually exclusive?
5. What real-world examples involve dependent probability?

Tip: When determining independence, always compare $P(A \cap B)$ with $P(A) \cdot P(B)$. If they are equal, the events are independent; otherwise, they are dependent.