difference between conditional probability and bayes rule I know the Bayes rule is derived from the conditional probability. But intuitively, what is the difference? The equation looks the same to me. The nominator is the joint probability and the denominator is the probability of the given outcome.
This is the conditional probability: $P(A∣B)=\frac{P(A \cap B)}{P(B)}$
This is the Bayes' rule: $P(A∣B)=\frac{P(B|A) * P(A)}{P(B)}$.
Isn't $P(B|A)*P(A)$ and $P(A \cap B)$ the same? When $A$ and $B$ are independent, there is no need to use the Bayes rule, right?
 A: OK, now that you have updated your question to include the two formulas:
$$P(A\mid B) = \frac{P(A\cap B)}{P(B)} ~~ \text{provided that }
 P(B) > 0, \tag{1}$$
is the definition of the conditional probability of $A$ given that
$B$ occurred.  Similarly,
$$P(B\mid A) = \frac{P(B\cap A)}{P(A)} =  \frac{P(A\cap B)}{P(A)} ~~ \text{provided that }
 P(A) > 0, \tag{2}$$
is the definition of the conditional probability of $B$ given that
$A$ occurred.  Now, it is true that it is a trivial matter to
substitute the value of $P(A\cap B)$ from $(2)$ into $(1)$ to
arrive at
$$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)} ~~ \text{provided that }
 P(A), P(B) > 0, \tag{3}$$
which is Bayes' formula but notice that Bayes's formula actually connects two different conditional probabilities $P(A\mid B)$
and $P(B\mid A)$, and is essentially a formula for "turning the
conditioning around". The Reverend Thomas Bayes referred to this
in terms of "inverse probability" and even today, there is
vigorous debate as to whether statistical inference should be
based on $P(B\mid A)$ or the inverse probability (called
the a posteriori or posterior probability).
It is undoubtedly as galling to you as it was to me when I first
discovered that Bayes' formula was just a trivial substitution of
$(2)$ into $(1)$. Perhaps if you have been born 250 years ago,
you (Note: the OP masqueraded under username AlphaBetaGamma when I wrote this answer but has since changed his username) could have made the substitution and then
people today would be talking about the AlphaBetaGamma formula and the
AlphaBetaGammian heresy and the Naive AlphaBetaGamma method$^*$ instead
of invoking Bayes' name everywhere. So
let me console you on your loss of fame by pointing out a different
version of Bayes' formula.  The Law of Total Probability says
that
$$P(B) = P(B\mid A)P(A) + P(B\mid A^c)P(A^c) \tag{4}$$
and using this, we can write $(3)$ as
$$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid A^c)P(A^c)}, \tag{5}$$
or more generally as
$$P(A_i\mid B) = \frac{P(B\mid A_i)P(A_i)}{P(B\mid A_1)P(A_1) + P(B\mid A_2)P(A_2) + \cdots + P(B\mid A_n)P(A_n)}, \tag{6}$$
where the posterior probability of a possible "cause" $A_i$ of a
"datum" $B$ is related to $P(B\mid A_i)$, the likelihood of the
observation $B$ when $A_i$ is the true hypothesis and $P(A_i)$, the prior probability
(horrors!) of the hypothesis $A_i$.

$^*$ There is a famous paper R. Alpher, H. Bethe, and
G. Gamow, "The Origin of Chemical Elements", Physical Review, April 1, 1948, that is commonly referred to as
the $\alpha\beta\gamma$ paper.
A: One way to intuitively think of Bayes' theorem is that when any one of these is easy to calculate
$$P(A∣B) ~~ \text{or } P(B∣A)$$
we can calculate the other one even though the other one seems to be bit hard at first
Consider an example, 
Here $$P(A∣B)$$ is say I have a curtain and I told you there is an animal behind the curtain and given it is a four legged animal what is the probability of that animal being dog ?
It is hard to find a probability for that.
But you can find the answer for $$P(B∣A)$$ 
What is the probability of a four legged animal behind the curtain and given that it is a dog, now it is easy to calculate  it could be nearly 1 and you plug in those values in the bayes theorem and you ll find the answer for $$P(A∣B)$$ that is the probability of the animal being a dog which at first was hard.
Now this is just an over simplified version where you can intuitively think why rearranging the formula could help us.
I hope this helps.
A: While converting P(A|B) to P(B|A) might be helpful in probability problems, we should take care not to imply causality. A large number of people with umbrellas (A) might indicate a high probability of rain (B), and rain (B) may equally indicate a high probability of umbrellas (A).  We might be able to argue that the rain causes the umbrellas (A -> B), but we cannot argue that the umbrellas cause he rain (B -> A).
