2.已知事件A,B满足P(AB)=P(A ̄B ̄ ) ,且P(A)=0.4,则P(B)=(C);
A.0.4　　B.0.5    C.0.6　　D.0.7


To solve this problem, we need to use the given probabilities and the relationship between them.

We are given:
- \( P(AB) = P(A^c B^c) \)
- \( P(A) = 0.4 \)

Let \( P(B) = p \).

We can start by expressing the given relationship in terms of probabilities:
\[ P(AB) = P(A^c B^c) \]

Using the properties of probabilities, we know that:
\[ P(A^c B^c) = 1 - P(A \cup B) \]

Using the inclusion-exclusion principle, we can write:
\[ P(A \cup B) = P(A) + P(B) - P(AB) \]

Therefore:
\[ P(A^c B^c) = 1 - [P(A) + P(B) - P(AB)] \]

Given that \( P(AB) = P(A^c B^c) \), we can set up the following equation:
\[ P(AB) = 1 - [P(A) + P(B) - P(AB)] \]

Plugging in \( P(A) = 0.4 \) and \( P(B) = p \), we get:
\[ P(AB) = 1 - [0.4 + p - P(AB)] \]

This simplifies to:
\[ P(AB) = 1 - 0.4 - p + P(AB) \]
\[ 0 = 0.6 - p \]
\[ p = 0.6 \]

Therefore, the probability \( P(B) \) is 0.6. 

The correct answer is:
C. 0.6

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you calculate the probability of the complement of an event?
2. What is the inclusion-exclusion principle in probability?
3. Can you explain the concept of conditional probability?
4. How do you find the probability of the intersection of two events?
5. What are some examples of mutually exclusive events?

**Tip:** When solving probability problems, always check if there are complementary events or use properties like inclusion-exclusion to simplify your calculations.

Learn how to solve a probability problem where you need to find the probability of event B given that the probability of the intersection of events A and B equals the probability of the complement of the intersection. This problem involves understanding basic probability concepts such as complement of an event and using the inclusion-exclusion principle. Suitable for high school students studying 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