(a)  If we have 2 events, A and B, that are mutually exclusive and we also know that P(A) = 0.40 and P(B) = 0.50. Find

(i)                 P(AUB)?

(ii)              P(AՌB)?

(iii)            P(A│B)?

Given that we have two events \(A\) and \(B\) that are **mutually exclusive**, and we know the probabilities \(P(A) = 0.40\) and \(P(B) = 0.50\), we can find the following probabilities:

### (i) \(P(A \cup B)\)
The probability of the union of \(A\) and \(B\), denoted as \(P(A \cup B)\), is the probability that either event \(A\) or event \(B\) occurs. Since \(A\) and \(B\) are mutually exclusive, they cannot occur together. Therefore, we can simply add their probabilities:

\[
P(A \cup B) = P(A) + P(B)
\]

Substitute the given values:

\[
P(A \cup B) = 0.40 + 0.50 = 0.90
\]

### (ii) \(P(A \cap B)\)
The probability of the intersection of \(A\) and \(B\), denoted as \(P(A \cap B)\), is the probability that both events \(A\) and \(B\) occur simultaneously. However, since \(A\) and \(B\) are mutually exclusive, the probability that both occur is zero:

\[
P(A \cap B) = 0
\]

### (iii) \(P(A \mid B)\)
The probability of \(A\) given \(B\), denoted as \(P(A \mid B)\), is the conditional probability that event \(A\) occurs given that event \(B\) has occurred. For mutually exclusive events, the occurrence of one event means the other cannot occur. Therefore:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Since \(P(A \cap B) = 0\) (because they are mutually exclusive):

\[
P(A \mid B) = \frac{0}{0.50} = 0
\]

### Summary of Results

1. \(P(A \cup B) = 0.90\)
2. \(P(A \cap B) = 0\)
3. \(P(A \mid B) = 0\)

Would you like to explore any of these concepts further or discuss related questions?

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### Relative Questions:
1. How would the probabilities change if \(A\) and \(B\) were not mutually exclusive?
2. Can mutually exclusive events be independent? Why or why not?
3. How would you calculate the probability of \(P(B \mid A)\) given different values?
4. What does it mean if \(P(A \mid B)\) is not zero, but \(A\) and \(B\) are still dependent events?
5. How would the scenario change if you introduced a third event \(C\) that is also mutually exclusive with \(A\) and \(B\)?

### Tip:
When working with mutually exclusive events, remember that their intersection is always zero, which simplifies many probability calculations.

Explore the probabilities of mutually exclusive events A and B with a detailed solution. Understand the concepts of probability theory, including conditional probabilities and the union of events. Suitable for undergraduate students studying probability.

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