probability?

Question

hi

i have the formula
P(A|B) = P(A n B) / P(B) (where n is intersection!)

but i am confused because isnt P(A n B) equal to P(A)P(B)
so we would get P(A)P(B) / P(B) so the P(B) would just cancel.

Am i right in thinking this?
thanks

thanks for your answers so far!

ok, so i have a value for P(A) and P(B) but they are not independent. Can / how do i work out P(A|B) from this knowledge?

Answer 1

The other answers are correct in saying that:
P(A n B) = P(A).P(B)
only when the events are independent.

As for the second part of your question, you don't have enough information to find out P(A|B) if you only have P(A) and P(B). To understand why, a couple of examples:

If A and B are mutually exclusive, P(A|B) (conditional probability of A given that B has occured) and P(B|A) are = 0, whatever values you have for P(A) and P(B).

If each event inevitably leads to the other, P(A|B) and P(B|A) always = 1, regardless of P(A) and P(B).

In other cases, the values will be somewhere in between!

You need more data to start with, that describes the relationship between event A and event B - you know that they are not independent, but how does the probability of each event depend on the other? This isn't a function of P(A) and P(B), you need information such as one or more of the conditional probabilities and/or the joint probability P(A n B), then you can calculate the unknown factors using the probability laws and Bayes' Theorem.

Hope that helps

Answer 2

P(A n B) = P(A)P(B) only when the events A and B are independed!

For example, if you throw a dice and denote
A... the number of dots on the dice is even
B... the number of dots on the dice is 2
then A,B are not independed.

You have to evaluate P(A n B) separately, or, if you know P(A|B) and P(B), use:
P(A n B)=P(A|B) P(B)

Answer 3

It is only true that P(AnB)=P(A)P(B) provided that A and B are independent events. Otherwise, it is false, and they do not cancel.

Answer 4

im not sure that kinda just confused me but yea i thnk u are right.