
*1 point*

For three events, A, B, and C, with P(C)>0*P*(*C*)>0, Which of the following is/are correct?

P(Ac∣C)*P*(*Ac*∣*C*)= 1 - P(A∣C)*P*(*A*∣*C*)

P(ϕ∣C)=0*P*(*ϕ*∣*C*)=0

P(A∣C)≤1*P*(*A*∣*C*)≤1

if A⊂B*A*⊂*B* then P(A∣C)≤P(B∣C)*P*(*A*∣*C*)≤*P*(*B*∣*C*)

Given the three events \( A \), \( B \), and \( C \) with \( P(C) > 0 \), let's analyze each statement to determine which are correct.

### 1. \( P(A^c \mid C) = 1 - P(A \mid C) \)
- **True**: This statement is correct. The conditional probability of the complement of \( A \) given \( C \) is equal to \( 1 \) minus the conditional probability of \( A \) given \( C \). Mathematically, this can be written as:
  \[
  P(A^c \mid C) = 1 - P(A \mid C)
  \]
  This follows directly from the definition of conditional probability and the fact that \( P(A) + P(A^c) = 1 \).

### 2. \( P(\emptyset \mid C) = 0 \)
- **True**: This statement is correct. The probability of the empty set given any event \( C \) is always 0. This is because the empty set \( \emptyset \) represents an impossible event, so \( P(\emptyset \mid C) = 0 \).

### 3. \( P(A \mid C) \leq 1 \)
- **True**: This statement is correct. The probability of any event \( A \) conditioned on \( C \) must lie between 0 and 1, inclusive. Therefore, \( P(A \mid C) \leq 1 \).

### 4. If \( A \subset B \) then \( P(A \mid C) \leq P(B \mid C) \)
- **True**: This statement is correct. If \( A \subset B \), then \( P(A \mid C) \) is less than or equal to \( P(B \mid C) \). This is because the probability of a subset (smaller event) occurring is less than or equal to the probability of the larger event occurring when conditioned on the same event \( C \).

### Conclusion:
- All four statements are correct.

Would you like more details or have any questions?

Here are some related questions:

1. What is the formula for conditional probability?
2. Can you prove the property \( P(A \mid C) + P(A^c \mid C) = 1 \) using basic probability rules?
3. What is the interpretation of \( P(A \mid C) \) in real-world scenarios?
4. How does the law of total probability apply when conditioning on an event \( C \)?
5. Can you provide examples where \( P(A \mid C) = P(A) \), meaning event \( C \) is independent of event \( A \)?

**Tip:** Remember that the conditional probability formula \( P(A \mid C) = \frac{P(A \cap C)}{P(C)} \) is only valid when \( P(C) > 0 \).

Explore the concepts of conditional probability with detailed explanations and examples. Learn about the complement rule, subset rule in probability, and how to apply these rules in real-world scenarios. Suitable for advanced high school students and anyone interested in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation