Conditional probability without conditional probability (yes, sounds odd) I am an amateur investigating about probability on my own. When I learned about conditional probabilities, I wanted to know what is the formula to calculate them, and I've found this:
P(A|B)=P(A and B)/P(B)
By the multiplication rule for joint probabilities, we have that:
P(A|B)=P(B|A).P(A)/P(B)
Ok, so this is Bayes rule right? Now, how in turn do I calculate P(B|A)? Isn't this circular? Because, it seems that, in order to calculate a conditional probability, I have to calculate another conditional probability, and so on. Is there a way of calculate conditional probability without pre-supposing conditional probability itself?
Thank you very much.
 A: It all depends on what you have, and what you want to know. This is rather a general mathematical thing than related to probabilities in particular. The formula
$$
  P(B|A)=\frac{P(A|B)\cdot P(B)}{P(A)}
$$
gives you $P(B|A)$ if you have $P(A|B)$, $P(B)$ and $P(A)$ (i.e. everything on the right hand side of the equation, that's what you always need to evaluate it).
Sometimes this is the case. But other times, you don't have $P(A|B)$, but instead only $P(A \text{ and } B)$ (often denoted $P(A,B)$). Then you might want to use this formula:
$$
  P(B|A)=\frac{P(A,B)}{P(A)}
$$
Do note that, since both give the same result, they should be the same formula. And indeed, we get the second version from the first by using $P(A|B)\cdot P(B) = P(A,B)$ (using your first formula).
So both equations say the same, but the individual expressions are "combined" differently. The second version does (in a sense) contain a conditional probability also, but merged in the combined ("joint") $P(A,B)$.
Therefore:


*

*If you have the "joint" probability $P(A,B)$ and the separate probability $P(A)$, you can get the conditional probability $P(B|A)$ by my second equation. You don't need to have any conditional probability at hand.

*If you have one conditional probability $P(A|B)$, then you can get the opposite conditional probability $P(B|A)$ directly by Bayes' rule (my first equation).

*Interestingly, you can also get $P(B|A)$ without having anything else that occurs in Bayes' rule. How? Consider measuring the body height of 1000 males (but no female). You can use this for example to establish $P(\text{height} > 1.70\, \mathrm{m} |  \text{sex = "male"})$. It's a perfectly valid conditional probability, and you didn't come across $P(\text{height} > 1.70\, \mathrm{m})$ or $P(\text{sex = "male"})$. That, unfortunately, also means that you can't use your statistics to find $P(\text{sex = "male"} | \text{height} > 1.70\, \mathrm{m})$, which would be a typical application if you wanted to create a classifier that guesses the sex of a human given the body height.
So the above formulas are only for computing conditional probabilities from things you might have available. They don't define them. They just state computation rules that are valid for them.
