# Decompostion of conditional probabilities

• torquerotates
In summary, under these conditions, P(B) = 1/2 x [(1/2)/(1/4)]xP(A|B) because the probability of an even number coming up on the die is 1/4 and the probability of a 1 or a 2 coming up is 1/2.
torquerotates
So If B=B1UB2U...UBn would Pr(A|B)=Pr(A|B1)Pr(B1)+Pr(A|B2)Pr(B2)+...+Pr(A|Bn)Pr(Bn)?

I haven't found a formula for this but it makes intuitive sense.

torquerotates said:
So If B=B1UB2U...UBn would Pr(A|B)=Pr(A|B1)Pr(B1)+Pr(A|B2)Pr(B2)+...+Pr(A|Bn)Pr(Bn)?

I haven't found a formula for this but it makes intuitive sense.

Hey torquerotates.

Before I give my take on the answer, do you assume that all the B1, B2, etc are disjoint? In other words is Bi AND Bj = NullSet for i != j?

Yes they are disjoint.

torquerotates said:
Yes they are disjoint.

For two events I get the following:

P(A|B1UB2) = P(A and (B1UB2))/P(B1UB2) = P((A and B1) U (A and B2))/P(B1UB2) = P(A and B1)/P(B1UB2) + P(A and B2)/P(B1UB2) - P(A and B1 and B2)/P(B1UB2)

= P(A and B1)/(P(B1) + P(B2)) + P(A and B1)/(P(B1) + P(B2)) - 0

= 1/P(B)[P(A and B1) + P(A and B2)]

= 1/P(B)[P(A and B1]P(B1)/P(B1) + P(A and B2)P(B2)/P(B2)]

= 1/P(B)[P(A|B1)P(B1) + P(A|B2)P(B2)] if we assume P(Bi) > 0 and is a valid probability.

If the above is right then your answer is wrong in general but right when B is the universal probability space.

Because your probability P(B) may not be one, it means that the above needs to be corrected using this information. Just for an example consider P(B1) = P(B2) = 1/4 and P(A|B1) = P(A|B2). But since 1/4[P(A|B1) + P(A|B2)] = 1/2P(A|B) for this example since P(B1) = P(B2) and since B1 and B2 are disjoint we have:

Then under these conditions we have:

1/P(B) x [P(A|B1)P(B1) + P(A|B2)P(B2)]
= 1/(1/2) x [P(A|B1)x1/4 + P(A|B1)x1/4]
=(1/4)/(1/2) x [P(A|B1) + P(A|B2)]
= 1/2 x [(1/2)/(1/4)]xP(A|B)
= 1/2 x 2 x P(A|B)
= P(A|B) for this example.

If we did not 'normalize' by P(B) we would not have gotten the right answer.

While I think the above is right, I would like others to check if they could just to make sure, but I'm only using a few identities which are basically distribution over sets and multiplication by 'x/x' terms for P(Bi)/P(Bi).

torquerotates said:
it makes intuitive sense.

Did you do any examples?

Let
A = the roll of a fair die produces an even number
B = the face that comes up is a 1 or a 2
B1 = the face that comes up is a 1
B2 = the face that comes up is a 2

## 1. What is decomposition of conditional probabilities?

Decomposition of conditional probabilities is a statistical technique used to break down a complex conditional probability into simpler and more manageable components. It involves breaking down a joint probability into a product of conditional probabilities, which can help in understanding the relationship between different variables.

## 2. Why is decomposition of conditional probabilities important in statistics?

Decomposition of conditional probabilities allows for a better understanding of the relationships and dependencies between different variables in a given dataset. It also helps in simplifying complex probability calculations and can be used to identify key factors that influence the overall probability.

## 3. How is decomposition of conditional probabilities different from marginal and joint probabilities?

Marginal probabilities represent the probability of a single event occurring, while joint probabilities represent the probability of multiple events occurring together. Decomposition of conditional probabilities breaks down a joint probability into a product of conditional probabilities, each representing the probability of a single event occurring given the occurrence of another event.

## 4. What are some common applications of decomposition of conditional probabilities?

Decomposition of conditional probabilities can be used in various fields, including finance, economics, and social sciences. Some common applications include analyzing consumer behavior, predicting stock market trends, and identifying risk factors in healthcare data.

## 5. Are there any limitations to using decomposition of conditional probabilities?

One limitation of decomposition of conditional probabilities is that it assumes independence between the variables being studied. In real-world situations, this may not always be the case, and the results of the decomposition may not accurately reflect the relationships between the variables. Additionally, the decomposition process can become complex and time-consuming for datasets with a large number of variables.

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