We are given that events $A$, $B$, and $C$ have certain properties, and we need to find the largest possible value for $P(A)$ under two different sets of conditions.

Part (a) – When $A$ is disjoint with both $B$ and $C$:

Key information:

• $P(B) = 0.5$
• $P(C) = 0.8$
• $P(B \cap C) = 0.45$
• $A \cap B = \emptyset$ (A and B are disjoint)
• $A \cap C = \emptyset$ (A and C are disjoint)

Since $A$ is disjoint with both $B$ and $C$, this means:

• $P(A \cap B) = 0$
• $P(A \cap C) = 0$

Thus, $A$ cannot overlap with either $B$ or $C$, so $P(A)$ must be less than or equal to the remaining portion of the sample space that is not occupied by $B \cup C$.

We can calculate the probability of $B \cup C$ as follows: $P(B \cup C) = P(B) + P(C) - P(B \cap C) = 0.5 + 0.8 - 0.45 = 0.85$ Since $B \cup C$ covers 85% of the sample space, the remaining portion is: $1 - P(B \cup C) = 1 - 0.85 = 0.15$ Therefore, the largest value that $P(A)$ can take is the remaining 0.15. Hence: $P(A) \leq 0.15$ The largest possible value of $P(A)$ in this case is 0.15.

Part (b) – When $A$ is disjoint with $B$ but not disjoint with $C$:

Key information:

• $P(A \cap B) = 0$ (A and B are disjoint)
• $A \cap C \neq \emptyset$ (A and C are not disjoint)

In this case, $A$ can overlap with $C$, but it must still be disjoint with $B$. To maximize $P(A)$, we want $A$ to include as much probability as possible. Since $A$ cannot overlap with $B$, we know that: $P(A \cap B) = 0$

However, $A$ can overlap with $C$, and $C$ has a probability of 0.8. The largest value for $P(A)$ would occur when $A$ includes all of the probability from $C$, along with any portion of the sample space that is not part of $B$.

Thus, $A$ can consist of:

• All of $C$, which has probability 0.8
• Any part of the sample space outside of $B$, since $A$ is disjoint with $B$.

Since $P(B) = 0.5$, the portion of the sample space not occupied by $B$ is $1 - P(B) = 0.5$. Therefore, $A$ can include up to 0.5 from the total sample space.

The largest value for $P(A)$ occurs when $A$ consists of all of $C$ (which has a probability of 0.8), but because $P(A)$ cannot overlap with $B$, we must subtract the intersection $P(B \cap C)$ from this maximum: $P(A) = P(C) = 0.8$ Hence, the largest possible value of $P(A)$ is 0.8.

Summary of Results:

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

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Would you like further details on any part of this problem?

Here are 5 related questions for further practice:

1. What is the probability of $B^c \cap C$?
2. If $A \cap B \neq \emptyset$, how does this affect $P(A)$?
3. Can two events be disjoint and independent? Explain.
4. How would you calculate $P(B \cup C^c)$?
5. What is the minimum value of $P(A)$ in part (b)?

Tip: When working with probabilities of combined events, always check if events are independent, disjoint, or overlapping, as these properties will guide your calculation strategy.