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.
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.
Chapter9: Sequences, Probability And Counting Theory
Section9.7: Probability
Problem 5SE: The union of two sets is defined as a set of elements that are present in at least one of the sets....
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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:
\( P((A \cap B)^c) = 1 - P(A \cap B) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F237b886a-f1e2-4141-9026-f97111290a4d%2Fe6f53c8e-1c6e-4bfa-9e7d-87c050bc9fe3%2Fap88dn4_processed.jpeg&w=3840&q=75)
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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