Suppose A and B are two events with probabilities: Find the following: a) P(ANB). b) P(A). c) P(Bº). P(Aª) = .40, P(B) = .25, P(AUB) = .75.

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**Question 6**

Suppose A and B are two events with probabilities:

\[ P(A^c) = 0.40, \, P(B) = 0.25, \, P(A \cup B) = 0.75. \]

Find the following:

a) \( P(A \cap B) \).

b) \( P(A) \).

c) \( P(B^c) \).

d) \( P((A \cap B)^c) \).

---

#### Explanation:

- \( P(A^c) \) represents the probability of the complement of event A.
- \( P(B) \) is the probability of event B.
- \( P(A \cup B) \) is the probability of the union of events A and B.

To find the required probabilities:
- **For (a)**: Use the formula for the intersection based on union:  
  \( P(A \cap B) = P(A) + P(B) - P(A \cup B) \).
- **For (b)**: Use the complement rule:  
  \( P(A) = 1 - P(A^c) \).
- **For (c)**: Again use the complement rule:  
  \( P(B^c) = 1 - P(B) \).
- **For (d)**: Use the complement rule on the intersection:  
  \( P((A \cap B)^c) = 1 - P(A \cap B) \).
Transcribed Image Text:**Question 6** Suppose A and B are two events with probabilities: \[ P(A^c) = 0.40, \, P(B) = 0.25, \, P(A \cup B) = 0.75. \] Find the following: a) \( P(A \cap B) \). b) \( P(A) \). c) \( P(B^c) \). d) \( P((A \cap B)^c) \). --- #### Explanation: - \( P(A^c) \) represents the probability of the complement of event A. - \( P(B) \) is the probability of event B. - \( P(A \cup B) \) is the probability of the union of events A and B. To find the required probabilities: - **For (a)**: Use the formula for the intersection based on union: \( P(A \cap B) = P(A) + P(B) - P(A \cup B) \). - **For (b)**: Use the complement rule: \( P(A) = 1 - P(A^c) \). - **For (c)**: Again use the complement rule: \( P(B^c) = 1 - P(B) \). - **For (d)**: Use the complement rule on the intersection: \( P((A \cap B)^c) = 1 - P(A \cap B) \).
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