Math Problem Statement

Show that P(A̅∩ B) = P(B) − P(A ∩ B).

Solution

To show that P(AB)=P(B)P(AB)P(A^\complement \cap B) = P(B) - P(A \cap B), we'll proceed with a step-by-step approach.

Definitions:

  • AA^\complement is the complement of AA, i.e., the event that AA does not occur.
  • ABA^\complement \cap B is the event where BB occurs but AA does not occur.
  • P(AB)P(A \cap B) is the probability that both AA and BB occur.

Step-by-step explanation:

We can break down event BB into two mutually exclusive events:

  1. ABA \cap B: The event where both AA and BB occur.
  2. ABA^\complement \cap B: The event where AA does not occur, but BB does.

Since ABA \cap B and ABA^\complement \cap B are mutually exclusive (they cannot happen at the same time), we can write:

B=(AB)(AB)B = (A \cap B) \cup (A^\complement \cap B)

Now, using the additive rule of probability for mutually exclusive events, we have:

P(B)=P(AB)+P(AB)P(B) = P(A \cap B) + P(A^\complement \cap B)

Rearranging this equation to isolate P(AB)P(A^\complement \cap B):

P(AB)=P(B)P(AB)P(A^\complement \cap B) = P(B) - P(A \cap B)

Thus, we have shown that:

P(AB)=P(B)P(AB)P(A^\complement \cap B) = P(B) - P(A \cap B)

Conclusion:

This proves the given equation.


Would you like more details or have any questions about the steps?

Here are some related questions for further exploration:

  1. What is the difference between the complement of an event and the intersection of two events?
  2. How does the addition rule of probabilities work when events are not mutually exclusive?
  3. What is the formula for the probability of the union of two events?
  4. How do we interpret the complement of an event in probability theory?
  5. Can this formula P(AB)=P(B)P(AB)P(A^\complement \cap B) = P(B) - P(A \cap B) be extended to more than two events?

Tip: When dealing with intersections and complements, always remember that the total probability of an event can often be decomposed into the sum of probabilities of mutually exclusive sub-events.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Set Theory
Complement of an Event
Intersection of Events

Formulas

P(A ∩ B)
P(A̅ ∩ B) = P(B) - P(A ∩ B)

Theorems

Additive Rule of Probability for Mutually Exclusive Events

Suitable Grade Level

Grades 11-12