Math Problem Statement
Show that P(A̅∩ B) = P(B) − P(A ∩ B).
Solution
To show that , we'll proceed with a step-by-step approach.
Definitions:
- is the complement of , i.e., the event that does not occur.
- is the event where occurs but does not occur.
- is the probability that both and occur.
Step-by-step explanation:
We can break down event into two mutually exclusive events:
- : The event where both and occur.
- : The event where does not occur, but does.
Since and are mutually exclusive (they cannot happen at the same time), we can write:
Now, using the additive rule of probability for mutually exclusive events, we have:
Rearranging this equation to isolate :
Thus, we have shown that:
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:
- What is the difference between the complement of an event and the intersection of two events?
- How does the addition rule of probabilities work when events are not mutually exclusive?
- What is the formula for the probability of the union of two events?
- How do we interpret the complement of an event in probability theory?
- Can this formula 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