what is exactly is the conditional probability I saw this this kind of definitions for conditional probability
Definition:
Conditional probability is the likelihood of an event occurring based on the occurrence of a previous event.
and some definitions follows like this
Definition:
$P(E|F) = P(EF)/P(F)$ (where $P(F) \neq 0$.) I saw the same formula in the first definition
I don't know which definition is defined first if 1 st definition is defined first then how do we came to the formula from the definition. or if 2nd definition (formula ) defined first than what does that formula actually says
in other words my question is what is the relation between the statement and formula
and I want to know this because that independent events are defined from this formula.
and one thing what does p(A|B) mean.
And additional question is what is this bayes theorem?
As I perceive it, you are unable to grasp how the formula $P(E|F) = P(EF)/P(F)$ (where $P(F) \neq0$ ) actually gives the conditional probability of $E$ occurring given that $F$ has ocurred.
Please go to the example and Venn Diagram of sets of creatures with two legs ($E$.say) and creatures that can fly.($F$)
The common region (intersection) is $EF$,
So if I ask what is the probability that the creature is two legged given that it can fly,notation $P(E|F)$ obviously it is the fraction of flyers who are two legged, ie $P(EF)/P(F)$
Thus $P(E|F) = P(EF)/P(F)$ expresses in a precise formula the conditional probability of $E$ given that $F$ has occurred.
If you look at the diagram again, you could also compute $P(F|E) = P(EF)/P(E)$,
so you can see that $P(EF) = P(E|F)*P(F) = P(F|E)*P(E)$
Bayes' Theorem converts one conditional probability to another, eg from P(E|F) to P(F|E)
A prime example is in disease testing. Suppose a COVID test has tested positive for you, P(Positive|Diseased), you want to know instead P(Diseased | Positive), you will need to use Bayes' Theorem, about you can read a bit more here