Conditional Probability Formula

[Avichal]

P(A/B) is defined to be P(A∩B)/P(B)

Why is this true?
When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram.
But for independent events, why is the formula correct? Any intuition or formal proof?

 

[mfb]

"Dependent events" always considers all cases, so independent events are a subset of those. If you understand the general case, using it in a special case should be no problem.
Anyway, it is a definition, asking about "correct or not" is meaningless.

 

[PeroK]

P(A/B) is defined to be P(A∩B)/P(B)

Why is this true?
When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram.
But for independent events, why is the formula correct? Any intuition or formal proof?

You can see this as follows:

$P(A \cap B) = P(B)P(A/B)$

Think about B happening "first".

If A and B both happen, then B must happen, then A must happen (given B has happened).

If A and B are independent, then $P(A/B) = P(A); \ P(A \cap B) = P(A)P(B)$ and the equation holds.

 

[chiro]

Hey Avichal.

The easiest way to convince yourself of it being true is to remember that if two events are independent, then one event happening will not in any way change the probability of another happening and vice-versa.

In mathematical notation this is defined as P(A|B) = P(A) given that B is a valid event (with a non-zero probability). If we use the definition of conditional probability along with this constraint we get:

P(A|B) = P(A and B)/P(B) = P(A) which implies
P(A and B) = P(A)*P(B) after multiplying both sides by P(B).