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.