To determine whether the events $A$ and $B$ are independent or dependent, we check if they satisfy the condition for independence:

$P(A|B) = P(A)$

If the conditional probability $P(A|B)$ is equal to the probability $P(A)$, then the events are independent. Otherwise, they are dependent.

Given:

• $P(A) = \frac{1}{9}$
• $P(B) = \frac{1}{3}$
• $P(A|B) = \frac{1}{2}$

Now, let's calculate the product $P(A) \times P(B)$ and compare it to $P(A|B) \times P(B)$.

If $P(A) \times P(B) = P(A \cap B)$, then the events are independent.

$P(A \cap B) = P(A|B) \times P(B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$

Now, compare $P(A \cap B)$ with $P(A) \times P(B)$:

$P(A) \times P(B) = \frac{1}{9} \times \frac{1}{3} = \frac{1}{27}$

Since $P(A \cap B) = \frac{1}{6} \neq \frac{1}{27}$, the events $A$ and $B$ are dependent.

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Would you like further details or have any questions?

Here are 5 related questions:

1. What is the definition of conditional probability?
2. How do we compute the joint probability of two events?
3. What is the difference between mutually exclusive and independent events?
4. How can we use a Venn diagram to illustrate dependent events?
5. How do dependent events affect the calculation of probabilities in a sequence of trials?

Tip: When events are dependent, the occurrence of one event changes the likelihood of the other event happening.