Let's solve the given problems step-by-step.

(a) Compute $P(A \text{ and } B)$.

We use the formula for the union of two events:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Plugging in the values provided:

$0.8 = 0.6 + 0.4 - P(A \cap B)$

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

$P(A \cap B) = 0.6 + 0.4 - 0.8 = 0.2$

So, $P(A \text{ and } B) = 0.2$.

(b) Are $A$ and $B$ mutually exclusive?

Two events are mutually exclusive if they cannot occur at the same time, i.e., $P(A \cap B) = 0$.

Since $P(A \cap B) = 0.2 \neq 0$, $A$ and $B$ are not mutually exclusive.

(c) Are $A$ and $B$ independent?

Two events $A$ and $B$ are independent if $P(A \cap B) = P(A) \times P(B)$.

Calculating $P(A) \times P(B)$:

$P(A) \times P(B) = 0.6 \times 0.4 = 0.24$

Since $P(A \cap B) = 0.2 \neq 0.24$, $A$ and $B$ are not independent.

Would you like more details or have any specific questions on these calculations?

Related Questions:

1. How do you calculate the probability of the complement of $A$ and $B$?
2. What is the probability of either $A$ or $B$ occurring, given their independence?
3. How would the result change if $P(A)$ was greater than 1?
4. What does it mean for events to be independent in real-life scenarios?
5. How do you verify mutual exclusivity if the events are part of a complex system?

Tip: Always remember that mutually exclusive events cannot happen simultaneously, while independent events have no effect on each other's occurrence.