Given 
P(A) = 0.3
 and 
P(B) = 0.5,
 do the following.
(a)
If A and B are mutually exclusive events, compute 
P(A or B).
(b)
If 
P(A and B) = 0.1,
 compute 
P(A or B).

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!

Given P(A) = 0.3 and P(B) = 0.5, do the following. (a) If A and B are mutually exclusive events, compute P(A or B). (b) If P(A and B) = 0.1, compute P(A or B).

Learn how to compute the probability of P(A or B) when events A and B are either mutually exclusive or not. This problem covers essential concepts of probability theory, including mutually exclusive events and the union of events. Suitable for students in Grades 9-11.

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