For events $A$ and $B$ to be independent, the probability of their intersection $P(A \cap B)$ must equal the product of their individual probabilities:

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

Given:

• $P(A) = 0.4$
• $P(A \cap B) = 0.2$

We can use the formula for independent events to find $P(B)$:

$0.2 = 0.4 \times P(B)$

Solving for $P(B)$:

$P(B) = \frac{0.2}{0.4}$ $P(B) = 0.5$

So, $P(B) = 0.5$.

Would you like more details or have any questions?

Here are some questions you might ask next:

1. How do you check if two events are independent?
2. Can events be both independent and mutually exclusive?
3. What are some real-life examples of independent events?
4. How do you calculate the probability of either event occurring for independent events?
5. What happens to the probabilities if events are dependent instead of independent?

Tip: When working with independent events, remember that the product of their individual probabilities gives the probability of their intersection. This helps in solving for unknown probabilities when some probabilities are known.