Events A and B are mutually exclusive. Suppose event A occurs with probability 0.2 and event B occurs with probability 0.39. Compute the following. (a)Compute the probability that A occurs or B does not occur (or both). (b)Compute the probability that either A occurs without B occurring or B occurs without A occurring.

Let's break down the problem into the two parts you're asked to compute:

### Part (a): Compute the probability that A occurs or B does not occur (or both).
This can be written as \( P(A \cup B^c) \).

#### Solution:
To compute \( P(A \cup B^c) \), we can use the following identity:
\[
P(A \cup B^c) = P(A) + P(B^c) - P(A \cap B^c)
\]
However, since \( A \) and \( B \) are mutually exclusive, \( A \cap B = \emptyset \), which means \( P(A \cap B) = 0 \). This also implies that \( A \cap B^c = A \).

So, \( P(A \cap B^c) = P(A) \).

Now, calculate:
\[
P(B^c) = 1 - P(B) = 1 - 0.39 = 0.61
\]

Substitute the values:
\[
P(A \cup B^c) = P(A) + P(B^c) - P(A \cap B^c) = 0.2 + 0.61 - 0.2 = 0.61
\]

Thus, the probability that \( A \) occurs or \( B \) does not occur (or both) is **0.61**.

### Part (b): Compute the probability that either A occurs without B occurring or B occurs without A occurring.
This can be interpreted as the probability of \( A \) occurring while \( B \) does not, or \( B \) occurring while \( A \) does not. This can be written as \( P(A \cap B^c) + P(B \cap A^c) \).

#### Solution:
Since \( A \) and \( B \) are mutually exclusive:
\[
P(A \cap B^c) = P(A) \quad \text{(since \( A \cap B = \emptyset \))}
\]
\[
P(B \cap A^c) = P(B) \quad \text{(since \( B \cap A = \emptyset \))}
\]

Thus, we add these probabilities:
\[
P(A \cap B^c) + P(B \cap A^c) = P(A) + P(B) = 0.2 + 0.39 = 0.59
\]

So, the probability that either \( A \) occurs without \( B \) occurring or \( B \) occurs without \( A \) occurring is **0.59**.

### Summary:
- **(a)** The probability that \( A \) occurs or \( B \) does not occur (or both) is **0.61**.
- **(b)** The probability that either \( A \) occurs without \( B \) occurring or \( B \) occurs without \( A \) occurring is **0.59**.

Would you like further details on these computations, or do you have any questions?

Here are some related questions you might explore:
1. What is the probability that neither A nor B occurs?
2. If A and B were not mutually exclusive, how would the calculations change?
3. What is the probability that both A and B occur?
4. What is the probability that either A or B occurs?
5. How would these probabilities change if the events were independent?
6. What is the significance of events being mutually exclusive in probability theory?
7. How do you compute conditional probabilities for mutually exclusive events?
8. How would the situation change if additional events C and D were introduced?

**Tip:** When dealing with mutually exclusive events, always remember that the probability of both events occurring simultaneously is zero.

Learn how to compute probabilities involving mutually exclusive events A and B with detailed step-by-step solutions. This problem covers calculating the probability of A occurring or B not occurring, and the probability of either A occurring without B or B occurring without A. Suitable for high school students studying probability theory.

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