To solve this problem, we need to understand how the probabilities of events $A$, $B$, and $C$ interact under different conditions.

Part (a): $A$ and $B$ are disjoint, and $A$ and $C$ are also disjoint.

• $A$ and $B$ being disjoint means $P(A \cap B) = 0$.
• $A$ and $C$ being disjoint means $P(A \cap C) = 0$.

Now, let's break this down:

• The probability of all events $A$, $B$, and $C$ must be less than or equal to 1.
• Since $P(B) = 0.5$ and $P(C) = 0.8$, the combined probability of $A$, $B$, and $C$ is constrained by: $P(A) + P(B) + P(C) - P(B \cap C) \leq 1$ Given that $P(B) = 0.5$, $P(C) = 0.8$, and $P(B \cap C) = 0.45$, substituting these values into the equation: $P(A) + 0.5 + 0.8 - 0.45 \leq 1$ Simplifying: $P(A) + 0.85 \leq 1$ So: $P(A) \leq 0.15$

Thus, the largest value that $P(A)$ can be in this case is 0.15.

Part (b): $A$ and $B$ are disjoint, but $A$ and $C$ are not disjoint.

• $A$ and $B$ are still disjoint, meaning $P(A \cap B) = 0$.
• However, now $A$ and $C$ are not disjoint, so $P(A \cap C)$ could be positive.

To maximize $P(A)$, we want to consider the case where $A$ and $C$ overlap as much as possible. The constraint is that the total probability still cannot exceed 1, so we need to find the upper bound by again considering: $P(A) + P(B) + P(C) - P(B \cap C) \leq 1$ Since $P(B) = 0.5$, $P(C) = 0.8$, and $P(B \cap C) = 0.45$, the equation becomes: $P(A) + 0.5 + 0.8 - 0.45 \leq 1$ Simplifying: $P(A) + 0.85 \leq 1$ So: $P(A) \leq 0.15$

However, because $A$ and $C$ can overlap, the portion of $P(A)$ that overlaps with $C$ can reduce the contribution of $C$ in the total probability. Therefore, to maximize $P(A)$, the entire $P(A)$ can overlap with $C$. In this case, $P(A)$ can be as large as $P(C)$, which is 0.8.

Thus, the largest value that $P(A)$ can be in this case is 0.8.

Summary:

• (a) The largest value of $P(A)$ when $A$ is disjoint from both $B$ and $C$ is 0.15.
• (b) The largest value of $P(A)$ when $A$ is disjoint from $B$, but not from $C$, is 0.8.

Would you like further clarification or more details on any step?

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Here are 5 related questions you might find useful:

1. How do we calculate the probability of disjoint events in general?
2. What is the inclusion-exclusion principle in probability?
3. How would the scenario change if the events were independent instead of disjoint?
4. What if $P(B \cap C) = 0$? How would this affect the answers in parts (a) and (b)?
5. How would we interpret conditional probabilities in the context of overlapping events?

Tip: Disjoint events (mutually exclusive) have no overlap, meaning their intersection is always zero. When calculating combined probabilities, remember this distinction for accurate results.