Are the events W and B independent? Provide proof.

I've already provided the solution for Question 4.2.4 in the previous message.

The events are NOT independent.

Here's a summary of the substantiation:
For independence, $P(W \cap B)$ must equal $P(W) \times P(B)$.
We found:

$$P(W \cap B) = \frac{50}{530} = \frac{5}{53}$$

And:

$$P(W) \times P(B) = \frac{340}{530} \times \frac{180}{530} = \frac{34}{53} \times \frac{18}{53} = \frac{612}{2809}$$

Since $\frac{5}{53} \neq \frac{612}{2809}$, the events are not independent.

The final answer is $\boxed{\text{No, the events are NOT independent. This is because } P(W \cap B) = \frac{5}{53} \text{ while } P(W) \times P(B) = \frac{34}{53} \times \frac{18}{53} = \frac{612}{2809}. \text{ Since } \frac{5}{53} \neq \frac{612}{2809}, \text{ the events are not independent.}}$.