Let's solve the given problems step by step.

(a) If $A$ and $B$ are mutually exclusive events:

If two events are mutually exclusive, they cannot happen at the same time, meaning $P(A \text{ and } B) = 0$. The formula to calculate the probability of $A \text{ or } B$ (the union of $A$ and $B$) is:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Since $P(A \text{ and } B) = 0$ (because the events are mutually exclusive), the formula simplifies to:

$P(A \text{ or } B) = P(A) + P(B)$

Now, substitute the given values:

$P(A \text{ or } B) = 0.3 + 0.5 = 0.8$

Thus, if $A$ and $B$ are mutually exclusive, $P(A \text{ or } B) = 0.8$.

(b) If $P(A \text{ and } B) = 0.1$:

Here, the events are not mutually exclusive, so we use the general formula for the probability of $A \text{ or } B$:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Substitute the given values:

$P(A \text{ or } B) = 0.3 + 0.5 - 0.1 = 0.7$

Thus, if $P(A \text{ and } B) = 0.1$, $P(A \text{ or } B) = 0.7$.

Summary:

• (a) If $A$ and $B$ are mutually exclusive, $P(A \text{ or } B) = 0.8$.
• (b) If $P(A \text{ and } B) = 0.1$, $P(A \text{ or } B) = 0.7$.

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Would you like further explanation or details on any part of this? Here are some related questions to explore:

1. What is the probability of both $A$ and $B$ occurring if they are mutually exclusive?
2. Can two events be mutually exclusive and independent at the same time?
3. How would the probabilities change if $P(A \text{ and } B)$ were 0.2 instead?
4. What is the complement of $P(A \text{ or } B)$?
5. How can you verify if two events are mutually exclusive using probability?

Tip: When events are mutually exclusive, their intersection (overlap) is always zero!