Understanding the Difference Between P (A, B) and P (B, A)

[spaghetti3451]

In proving Bayes' Theorem,

we use the following two statements.

P (A, B) = P (A|B) P (B)
P (B, A) = P (B|A) P (A).

I am wondering what's the difference between P (A, B) and P (B, A).

Any takers?

 

[Stephen Tashi]

There is no difference if by P(A,B) means the probability of the event "A and B". The event "A and B" is the same as the event "B and A".

 

[spaghetti3451]

Thanks!

 

[SW VandeCarr]

In proving Bayes' Theorem,

we use the following two statements.

P (A, B) = P (A|B) P (B)
P (B, A) = P (B|A) P (A).

I am wondering what's the difference between P (A, B) and P (B, A).

Any takers?

P(A,B) is functional notation which is to be defined such as in $$f(x,y)= 6x + y^2$$ for example. The order of variables in the argument doesn't usually matter unless specifically stated.

You've defined it in terms of probabilities two ways which can be written:

$$P(A\cap B)$$ and $$P(B \cap A)$$

They are the same but not because P(A,B) means P(A^B). P(A,B) is simply a function which is to be defined.

 

[blue_raver22]

P (A, B) and P (B, A) represent different conditional probabilities. P (A, B) represents the probability of both event A and event B occurring together, while P (B, A) represents the probability of both event B and event A occurring together. In other words, the order of the events in the notation matters.

In Bayes' Theorem, we use both P (A, B) and P (B, A) to calculate the probability of event A given that event B has occurred. This is represented by P (A|B). The first statement, P (A, B) = P (A|B) P (B), represents the joint probability of A and B, which is the probability of both events occurring together. The second statement, P (B, A) = P (B|A) P (A), represents the conditional probability of B given A, multiplied by the probability of A.

In order to fully understand Bayes' Theorem and apply it correctly, it is important to understand the difference between P (A, B) and P (B, A). Both represent different aspects of the conditional probability and are essential in the calculation of the final probability.