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.