Let A, B, and C be three events. We know that P (B) = 0.5, P (C) = 0.8,
and P (B ∩ C) = 0.45.
(a) If A and B are disjoint, and A and C are also disjoint, what is the
largest value that P (A) can be?
(b) If instead, A and B are disjoint, but A and C are not disjoint, what is
the largest value that P (A) can be?

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**.

---

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.

Let A, B, and C be three events. We know that P(B) = 0.5, P(C) = 0.8, and P(B ∩ C) = 0.45. (a) If A and B are disjoint, and A and C are also disjoint, what is the largest value that P(A) can be? (b) If instead, A and B are disjoint, but A and C are not disjoint, what is the largest value that P(A) can be?

Solve a probability problem involving disjoint and non-disjoint events A, B, and C. Learn how to calculate the largest value of P(A) based on given probabilities. This problem covers key probability concepts such as the addition rule for probability and the disjointness property. Suitable for students in Grades 10-12.

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