Proof of this probability equation

In summary, the conversation is discussing the proof of conditional probability for disjoint events. The formula is P(B1, B2, ... | A) = P(B1 | A) + P(B2 | A) + ... and the person is struggling to prove it with only the given equations P(A) > 0 and P(B ∩ A) ≥ 0. However, the other person suggests using the rule that if A and B are disjoint, then P(A∪B) = P(A) + P(B) to solve the problem. The final solution involves breaking down the intersection of events and using the formula P(B1, B2, ... | A) = P(B1 | A) + P(B2
  • #1
zeion
466
1

Homework Statement



Hi,

I don't understand the proof of this for conditional probability:

If B1, B2, ... are disjoint,
then P(B1, B2, B3, ... | A) = P(B1 | A) + P(B2 | A) + ...

From my notes I only have P(A) > 0 , P(B ∩ A) ≥ 0
but I don't see how those help me prove it..

Homework Equations





The Attempt at a Solution



P(B1, B2, ... | A) = P(B1, B2, B3, ... ∩ A) / P(A)

How do I get +'s out?
 
Physics news on Phys.org
  • #2
Well, you are right- if you have only those two rules to work with, you cannot possibly do it. But it seems to me that whoever gave you this exercise expected you to know more- specifically, that if A and B are disjoint, then [itex]P(A\cup B)= P(A)+ P(B)[/itex].
 
  • #3
Ok so I have:

P(B1, B2, ... | A)
= P(B1, B2, ... ∩ A) / P(A)
= (1/P(A)) (P(B1, B2, ... ∩ A)
= ?
= (1/P(A)) (P(B1 ∩ A) + P(B2 ∩ A) + ... )
= [P(B1 ∩ A) / P(A)] + [P(B2 ∩ A) / P(A)] + ...
= P(B1 | A) + P(B2 | A) + ...


how can I show that P(B1, B2, ... ∩ A) = P(B1 ∩ A) + P(B2 ∩ A) + ... ?
 

FAQ: Proof of this probability equation

What is "Proof of this probability equation"?

"Proof of this probability equation" refers to the mathematical process used to demonstrate that a probability equation is true. It involves using logical reasoning, mathematical principles, and evidence to show that the equation accurately represents the likelihood of an event occurring.

Why is "Proof of this probability equation" important?

"Proof of this probability equation" is important because it allows us to have confidence in the accuracy of a probability equation. It ensures that the equation is based on sound mathematical principles and can be relied upon to make informed decisions and predictions.

How is "Proof of this probability equation" different from a regular proof?

In a regular proof, the goal is to show that a statement or equation is true in all cases. However, in "Proof of this probability equation", the goal is to demonstrate that a probability equation is true only for a specific set of conditions or assumptions. This is because probability is based on chance and uncertainty, rather than absolute truth.

What are the key steps involved in "Proof of this probability equation"?

The key steps in "Proof of this probability equation" include defining the assumptions and conditions of the probability equation, using logical reasoning to manipulate the equation, and providing evidence to support the validity of the equation. This may involve using mathematical principles, empirical data, or statistical analysis.

Can "Proof of this probability equation" be done for any probability equation?

Yes, "Proof of this probability equation" can be done for any probability equation as long as the assumptions and conditions are clearly defined. However, the complexity of the equation and the availability of evidence may vary, making the process more challenging for some equations than others.

Similar threads

Replies
6
Views
5K
Replies
4
Views
2K
Replies
14
Views
2K
Replies
7
Views
1K
Replies
29
Views
3K
Replies
9
Views
2K
Back
Top